Open Access

Artin L-functions of small conductor

Research in Number Theory20173:16

DOI: 10.1007/s40993-017-0079-5

Received: 19 December 2016

Accepted: 21 March 2017

Published: 10 July 2017

Abstract

We study the problem of finding the Artin L-functions with the smallest conductor for a given Galois type. We adapt standard analytic techniques to our novel situation of fixed Galois type and obtain much improved lower bounds on the smallest conductor. For small Galois types we use complete tables of number fields to determine the actual smallest conductor.

Keywords

Artin representation L-function Number field Conductor

1 Overview

Artin L-functions \(L(\mathcal {X},s)\) are remarkable analytic objects built from number fields. Let \(\overline{\mathbf {Q}}\) be the algebraic closure of the rational number field \(\mathbf {Q}\) inside the field of complex numbers \(\mathbf {C}\). Then Artin L-functions are indexed by continuous complex characters \(\mathcal {X}\) of the absolute Galois group \({\mathbb {G}}= {{\mathrm{Gal}}}(\overline{\mathbf {Q}}/\mathbf {Q})\), with the unital character 1 giving the Riemann zeta function \(L(1,s) = \zeta (s)\). An important problem in modern number theory is to obtain a fuller understanding of these higher analogs of the Riemann zeta function. The analogy is expected to be very tight: all Artin L-functions are expected by the Artin conjecture to be entire except perhaps for a pole at \(s=1\); they are all expected to satisfy the Riemann hypothesis that all zeros with \(\text{ Re }(s) \in (0,1)\) satisfy \(\text{ Re }(s)=1/2\).

The two most basic invariants of an Artin L-function \(L(\mathcal {X},s)\) are defined via the two explicit elements of \({\mathbb {G}}\), the identity e and the complex conjugation element \(\sigma \). These invariants are the degree \(n = \mathcal {X}(e)\) and the signature \(r = \mathcal {X}(\sigma )\) respectively. A measure of the complexity of \(L(\mathcal {X},s)\) is its conductor \(D \in \mathbf {Z}_{\ge 1}\), which can be computed from the discriminants of related number fields. It is best for purposes such as ours to focus instead on the root conductor \(\delta = D^{1/n}\).

In this paper, we aim to find the simplest Artin L-functions exhibiting a given Galois-theoretic behavior. To be more precise, consider triples \((G,c,\chi )\) consisting of a finite group G, an involution \(c \in G\), possibly trivial, and a faithful character \(\chi \). We say that \(\mathcal {X}\) has Galois type \((G,c,\chi )\) if there is a surjection \(h : {\mathbb {G}}\rightarrow G\) with \(h(\sigma ) = c\), and \(\mathcal {X}= \chi \circ h\). Let \(\mathcal {L}(G,c,\chi )\) be the set of L-functions of type \((G,c,\chi )\), and let \(\mathcal {L}(G,c,\chi ;B)\) be the subset consisting of L-functions with root conductor at most B. Two natural problems for any given Galois type \((G,c,\chi )\) are
1:: 

Use known and the above conjectured properties of L-functions to obtain a lower bound \(\mathfrak {d}(G,c,\chi )\) on the root conductors of L-functions in \(\mathcal {L}(G,c,\chi )\).

2:: 

Explicitly identify the sets \(\mathcal {L}(G,c,\chi ;B)\) with B as large as possible.

This paper gives answers to both problems, although for brevity we often fix only \((G,\chi )\) and work instead with the sets \(\mathcal {L}(G,\chi ;B) := \cup _c \mathcal {L}(G,c,\chi ;B)\).

There is a large literature on a special case of the situation we study. Namely let \((G,c,\phi )\) be a Galois type where \(\phi \) is the character of a transitive permutation representation of G. Then the set \(\mathcal {L}(G,c,\phi ;B)\) is exactly the set of Dedekind zeta functions \(\zeta (K,s)\) arising from a corresponding set \(\mathcal {K}(G,c,\phi ;B)\) of arithmetic equivalence classes of number fields. In this context, root conductors are just root discriminants, and lower bounds date back to Minkowski’s work on the geometry of numbers. Use of Dedekind zeta functions as in 1 above began with work of Odlyzko [1719], Serre [29], Poitou [26, 27], and Martinet [16]. Extensive responses to 2 came shortly thereafter, with papers often focusing on a single degree \(n=\phi (e)\). Early work for quartics, quintics, sextics, and septics include respectively [24, 7, 13, 2123, 25, 31]. Further results towards 2 in higher degrees are extractable from the websites associated to [10, 12, 14].

The full situation that we are studying here was identified clearly by Odlyzko in [20], who responded to 1 with a general lower bound. However this more general case of Artin L-functions has almost no subsequent presence in the literature. A noticeable exception is a recent paper of Pizarro-Madariaga [24], who improved on Odlyzko’s results on 1. A novelty of our paper is the separation into Galois types. For many Galois types this separation allows us to go considerably further on 1. This paper is also the first systematic study of 2 beyond the case of number fields.

Sections 2 and 3 review background on Artin L-functions and tools used to bound their conductors. Sections 4, 5 and 6 form the new material on the lower bound problem 1, while Sects. 7, 8, and 9 focus on the tabulation problem 2. Finally, Sect. 10 returns to 1 and considers asymptotic lower bounds for root conductors of Artin L-functions in certain families. In regard to 1, Fig. 3 and Corollary 10.1 give a quick indication of how our type-based lower bounds compare with the earlier degree-based lower bounds. In regard to both 1 and 2, Tables 3, 4, 5, 6 and 7 show how the new lower bounds compare with actual first conductors for many types.

Artin L-functions have recently become much more computationally accessible through a package implemented in Magma [1] by Tim Dokchitser. Thousands are now collected in a section on the LMFDB [14]. The present work increases our understanding of all this information in several ways, including by providing completeness certificates for certain ranges.

2 Artin L-functions

In this section we provide some background. An important point is that our problems allow us to restrict consideration to Artin characters which take rational values only. In this setting, Artin L-functions can be expressed as products and quotients of roots of Dedekind zeta functions, minimizing the background needed. General references on Artin L-functions include [15, 28].

2.1 Number fields

A number field K has many invariants relevant for our study. First of all, there is the degree \(n = [K{:}\mathbf {Q}]\). The other invariants we need are local in that they are associated with a place v of \(\mathbf {Q}\) and can be read off from the corresponding completed algebra \(K_v = K \otimes \mathbf {Q}_v\), but not from other completions. For \(v=\infty \), the complete invariant is the signature r, defined by \(K_\infty \cong \mathbf {R}^{r} \times \mathbf {C}^{(n-r)/2}\). It is more convenient sometimes to work with the eigenspace dimensions for complex conjugation, \(a = (n+r)/2\) and \(b = (n-r)/2\). For an ultrametric place \(v=p\), the full list of invariants is complicated. The most basic one is the positive integer \(D_p = p^{c_p}\) generating the discriminant ideal of \(K_p/\mathbf {Q}_p\). We package the \(D_p\) into the single invariant \(D = \prod _p D_p \in \mathbf {Z}_{\ge 1}\), the absolute discriminant of K.

2.2 Dedekind zeta functions

Associated with a number field is its Dedekind zeta function
$$\begin{aligned} \zeta (K,s) = \prod _{p} \frac{1}{P_p(p^{-s})} = \sum _{m=1}^\infty \frac{a_m}{m^s}. \end{aligned}$$
(2.1)
Here the polynomial \(P_p(x) \in \mathbf {Z}[x]\) is a p-adic invariant. It has degree \(\le n\) with equality if and only if \(D_p=1\). The integer \(a_m\) is the number of ideals of index m in the ring of integers \(\mathcal {O}_K\).

2.3 Analytic properties of Dedekind zeta functions

Let \(\Gamma _\mathbf {R}(s) = \pi ^{-s/2} \Gamma (s/2)\), where \(\Gamma (s) = \int _0^\infty x^{s-1} e^{-x} dx\) is the standard gamma function. Let
$$\begin{aligned} \widehat{\zeta }(K,s) = D^{s/2} \Gamma _\mathbf {R}\left( s\right) ^a \Gamma _\mathbf {R}\left( s+1\right) ^b \zeta (K,s). \end{aligned}$$
(2.2)
Then this completed Dedekind zeta function \(\widehat{\zeta }(K,s)\) meromorphically continues to the whole complex plane, with simple poles at \(s=0\) and \(s=1\) being its only singularities. It satisfies the functional equation \(\widehat{\zeta }(K,s) = \widehat{\zeta }(K,1-s)\).

2.4 Permutation characters

We recall from the introduction that throughout this paper we are taking \(\overline{\mathbf {Q}}\) to be the algebraic closure of \(\mathbf {Q}\) in \(\mathbf {C}\) and \({\mathbb {G}}= {{\mathrm{Gal}}}(\overline{\mathbf {Q}}/\mathbf {Q})\) its absolute Galois group. A degree n number field K then corresponds to the transitive n-element \({\mathbb {G}}\)-set \(X = \text{ Hom }(K,\overline{\mathbf {Q}})\). A number field thus has a permutation character \(\Phi = \Phi _K = \Phi _X\) with \(\Phi (e)=n\). Also signature has the character-theoretic interpretation \(\Phi (\sigma ) = r\), where \(\sigma \) as before is the complex conjugation element.

2.5 General characters and Artin L-functions

Let \(\mathcal {X}\) be a character of \({\mathbb {G}}\). Then one has an associated Artin L-function \(L(\mathcal {X},s)\) and conductor \(D_\mathcal {X}\), agreeing with the Dedekind zeta function \(\zeta (K,s)\) and the discriminant \(D_K\) if \(\mathcal {X}\) is the permutation character of K. The function \(L(\mathcal {X},s)\) has both an Euler product and Dirichlet series representation as in (2.1). In general, if \(\Phi = \sum _\mathcal {X}m_{\mathcal {X}} \mathcal {X}\) then
$$\begin{aligned} L(\Phi ,s)&= \prod _{\mathcal {X}} L(\mathcal {X},s)^{m_{\mathcal {X}}} \quad D_\Phi = \prod _{\mathcal {X}} D_\mathcal {X}^{m_{\mathcal {X}}}. \end{aligned}$$
(2.3)
One is often interested in (2.3) where the \(\mathcal {X}\) are irreducible characters.

For a finite set of primes S, let \(\overline{\mathbf {Q}}_S\) be the compositum of all number fields in \(\overline{\mathbf {Q}}\) with discriminant divisible only by primes in S. Let \({\mathbb {G}}_S = {{\mathrm{Gal}}}(\overline{\mathbf {Q}}_S/\mathbf {Q})\) be the corresponding quotient of \({\mathbb {G}}\). Then for primes \(p \not \in S\) one has a well-defined Frobenius conjugacy class \(\text{ Fr }_p\) in \({\mathbb {G}}_S\). The local factor \(P_p(x)\) in (2.1) is the characteristic polynomial \(\det (1 - \rho (\text{ Fr }_p) x)\), where \(\rho \) is a representation with character \(\mathcal {X}\).

2.6 Relations with other objects

Artin L-functions of degree 1 are exactly Dirichlet L-functions, so that \(\mathcal {X}\) can be identified with a faithful character of the quotient group \((\mathbf {Z}/D\mathbf {Z})^\times \) of \({\mathbb {G}}\), with D the conductor of \(\mathcal {X}\). Artin L-functions coming from irreducible degree 2 characters and conductor D are expected to come from cuspidal modular forms on \(\Gamma _1(D)\), holomorphic if \(r=0\) and nonholomorphic otherwise. This expectation is proved in all cases, except for those with \(r = \pm 2\) and projective image the nonsolvable group \(A_5\). In general, to understand how an Artin L-function \(L(\mathcal {X},s)\) qualitatively relates to other objects, one needs to understand its Galois theory, including the placement of complex conjugation; in other words, one needs to identify its Galois type. To be more quantitative, one brings in the conductor.

2.7 Analytic properties of Artin L-functions

An Artin L-function has a meromorphic continuation and functional equation, although each with an extra complication in comparison with the special case of Dedekind zeta functions. For the meromorphic continuation, the behavior at \(s=1\) is known: the pole order is the multiplicity \((1,\mathcal {X})\) of 1 in \(\mathcal {X}\). The complication is that one has poor control over other possible poles. The Artin conjecture for \(\mathcal {X}\) says however that there are no poles other than \(s=1\).

The completed L-function
$$\begin{aligned} \widehat{L}(\mathcal {X},s) = D_\mathcal {X}^{s/2} \Gamma _\mathbf {R}(s)^a \Gamma _\mathbf {R}(s+1)^b L(\mathcal {X},s) \end{aligned}$$
satisfies the functional equation
$$\begin{aligned} \widehat{L}(\mathcal {X},1-s) = w \widehat{L}(\overline{\mathcal {X}},s) \end{aligned}$$
with root number w. Irreducible characters of any compact group come in three types, orthogonal, non-real, and symplectic. The type is identified by the Frobenius-Schur indicator, calculated with respect to the Haar probability measure dg:
$$\begin{aligned} FS(\chi ) = \int _{G} \chi (g^2) \, dg \in \{-1,0,1\}. \end{aligned}$$
For orthogonal characters \(\mathcal {X}\) of \({\mathbb {G}}\), one has \(\mathcal {X}= \overline{\mathcal {X}}\) and moreover \(w = 1\). The complication in comparison with permutation characters is that for the other two types, the root number w is not necessarily 1. For symplectic characters, \(\mathcal {X}= \overline{\mathcal {X}}\) and w can be either of the two possibilities 1 or \(-1\). For non-real characters, \(\mathcal {X}\ne \overline{\mathcal {X}}\) and w is some algebraic number of norm 1.

Recall from the introduction that an Artin L-function is said to satisfy the Riemann hypothesis if all its zeros in the critical strip \(0<\text{ Re }(s)<1\) are actually on the critical line \(\text{ Re }(s)= 1/2\). We will be using the Riemann hypothesis through Lemma 3.1. If we replaced the function (3.1) with the appropriately scaled version of (5.17) from [24], then our lower bounds would be only conditional on the Artin conjecture, which is completely known for some Galois types \((G,c,\chi )\). However the bounds obtained would be much smaller, and the comparison with first conductors as presented in Tables 3, 4, 5, 6 and 7 below would be less interesting.

2.8 Rational characters and rational Artin L-functions

The abelianized Galois group \({\mathbb {G}}^\mathrm{ab}\) acts on continuous complex characters of profinite groups through its action on their values. If \(\mathcal {X}'\) and \(\mathcal {X}''\) are conjugate via this action then their conductors agree:
$$\begin{aligned} D_{\mathcal {X}'} = D_{\mathcal {X}''}. \end{aligned}$$
(2.4)
Our study is simplified by this equality because it allows us to study a given irreducible character \(\mathcal {X}'\) by studying instead the rational character \(\mathcal {X}\) obtained by summing its conjugates.
By the Artin induction theorem [6, Prop. 13.2], a rational character \(\mathcal {X}\) can be expressed as a rational linear combination of permutation characters:
$$\begin{aligned} \mathcal {X}= \sum k_\Phi \Phi . \end{aligned}$$
(2.5)
For general characters \(\mathcal {X}'\), computing the Frobenius traces \(a_p = \mathcal {X}'(\text{ Fr }_p)\) requires the results of [5]. Similarly the computation of bad Euler factors and the root number w present difficulties. For Frobenius traces and bad Euler factors, these complications are not present for rational characters \(\mathcal {X}\) because of (2.5).

3 Signature-based analytic lower bounds

Here and in the next section we aim to be brief, with the main point being to explain how type-based lower bounds are usually much larger than signature-based lower bounds. We employ the standard framework for establishing lower bounds for conductors and discriminants, namely Weil’s explicit formula. General references for the material in this section are [20, 24].

3.1 Basic quantities

The theory allows general test functions that satisfy certain axioms. We work only with a function introduced by Odlyzko (see [27, (9)]),
$$\begin{aligned} f(x) = \left\{ \begin{array}{ll} {\displaystyle (1-x) \cos (\pi x) + \frac{\sin (\pi x)}{\pi }}, &{} \text{ if } 0 \le x \le 1\text{, } \\ 0, &{} \text{ if } x>1. \end{array} \right. \end{aligned}$$
(3.1)
For \(z \in [0,\infty )\) let
$$\begin{aligned} N(z)&= \log (\pi ) + \int _0^{\infty } \frac{e^{-x/4}+e^{-3x/4}}{2(1-e^{-x})} f(x/(2z)) - \frac{e^{-x}}{x} \,dx \\&= \gamma + \log (8\pi ) + \int _0^\infty \frac{f(x/z)-1}{2\sinh (x/2)} \, dx \\&= \gamma +\log (8\pi ) + \int _0^z \frac{f(x/z)-1}{2\sinh (x/2)} \, dx -\log \left( \frac{e^{z/2}+1}{e^{z/2}-1} \right) , \\ R(z)&= \int _0^{\infty }\frac{e^{-x/4}-e^{-3x/4}}{2(1-e^{-x})} f(x/(2z))\, dx \\&= \int _0^z \frac{f(x/z)}{2\cosh (x/2)}\, dx, \\ P(z)&= 4 \int _0^{\infty } f(x/z) \cosh (x/2)\, dx \\&= \frac{256 \pi ^2 z \cosh ^2(z/4)}{(z^2+4\pi ^2)^2}. \end{aligned}$$
The simplifications in the integrals for N(z) and R(z) are fairly standard and apply to any test function, with the exception of the final steps which make use of the support for f(x). Evaluation of P(z) depends on the choice of f(x). The integrals for N(z) and R(z) cannot be evaluated in closed form like the third, but, as indicated in [27, §2], they do have simple limits \(N(\infty ) = \log (8 \pi ) + \gamma \) and \(R(\infty ) = \pi /2\) as \(z \rightarrow \infty \). Here \(\gamma \approx 0.5772\) is the Euler \(\gamma \)-constant. The constants \(\Omega = e^{N(\infty )} \approx 44.7632\) and \(e^{R(\infty )} \approx 4.8105\), as well as their product \(\Theta = e^{N(\infty )+R(\infty )} \approx 215.3325\), will play important roles in the sequel.

3.2 The quantity M(nru)

Consider triples (nru) of real numbers with n and u positive and \(r \in [-n,n]\). For such a triple, define
$$\begin{aligned} M(n,r,u) = \text{ Max }_z \left( \exp \left( N(z) + \frac{r}{n} R(z) - \frac{u}{n} P(z) \right) \right) . \end{aligned}$$
It is clear that \(M(n,r,u) = M(n/u,r/u,1)\). Accordingly we regard \(u=1\) as the essential case and abbreviate \(M(n,r)=M(n,r,1)\). For fixed \(\epsilon \in [0,1]\) and \(u>0\), one has the asymptotic behavior
$$\begin{aligned} \lim _{n \rightarrow \infty } M(n,\epsilon n) = \Omega ^{1-\epsilon } \Theta ^{\epsilon } \approx 44.7632^{1-\epsilon } 215.3325^{\epsilon }. \end{aligned}$$
(3.2)
Figure 1 gives one a feel for the fundamental function M(nr). Particularly important are the univariate functions M(n, 0) and M(nn) corresponding to the left and right edges of this figure.
Fig. 1

A contour plot of \(M(n,\epsilon n)\) in the window \([0,1] \times \) [1,1,000,000] of the \(\epsilon \)-n plane, with a vertical logarithmic scale and contours at 2, 4, 6, 8, \(\mathbf{10}\), ..., 170, 172, 174, 176. Some limits for \(n \rightarrow \infty \) are shown on the upper boundary

3.3 Lower bounds for root discriminants

Suppose that \(\Phi \) is a nonzero Artin character which takes real values only. We say that \(\Phi \) is nonnegative if
$$\begin{aligned} \Phi (g) \ge 0 \quad {\text { for all }}g \in {\mathbb {G}}. \end{aligned}$$
(3.3)
This nonnegativity ensures that the inner product \((\Phi ,1)\) of \(\Phi \) with the unital character 1 is positive. A central result of the theory, a special case of the statement in [27, (7)], then serves us as a lemma.

Lemma 3.1

The lower bound
$$\begin{aligned} \delta _{\Phi } \ge M(n,r,u). \end{aligned}$$
is valid for all nonnegative characters \(\Phi \) with \((\Phi (e),\Phi (\sigma ),(\Phi ,1)) = (n,r,u)\) and \(L(\Phi ,s)\) satisfying the Artin conjecture and the Riemann hypothesis.

If \(\Phi \) is a permutation character, then the nonnegativity condition (3.3) is automatically satisfied. This makes the application of the analytic theory to lower bounds of root discriminants of fields relatively straightforward.

3.4 Lower bounds for general Artin conductors

To pass from nonnegative characters to general characters, the classical method uses the following lemma.

Lemma 3.2

[20] The conductor relation
$$\begin{aligned} \delta _\mathcal {X}\ge \delta _{\Phi }^{n/(2n-2)} \end{aligned}$$
(3.4)
holds for any degree n character \(\mathcal {X}\) and its absolute square \(\Phi = |\mathcal {X}|^2\).

A proof of this lemma from first principles is given in [20].

Combining Lemma 3.1 with Lemma 3.2 one gets the following immediate consequence

Theorem 3.3

The lower bound
$$\begin{aligned} \delta _\mathcal {X}\ge M(n^2,r^2,w)^{n/(2n-2)} \end{aligned}$$
is valid for all characters \(\mathcal {X}\) with \((\mathcal {X}(e),\mathcal {X}(\sigma ),(\mathcal {X},\overline{\mathcal {X}})) = (n,r,w)\) such that \(L(|\mathcal {X}|^2,s)\) satisfies the Artin conjecture and the Riemann hypothesis.
This theorem is essentially the main result in the literature on lower bounds for Artin conductors. It appears in [20, 24] with the right side replaced by explicit bounds. For fixed \(\epsilon \in [-1,1]\) and \(w>0\), one has the asymptotic behavior
$$\begin{aligned} \lim _{n \rightarrow \infty } M(n^2,\epsilon ^2 n^2,w) = \Omega ^{(1-\epsilon ^2)/2} \Theta ^{\epsilon ^2/2} \approx 6.6905^{1-\epsilon ^2} 14.6742^{\epsilon ^2}. \end{aligned}$$
(3.5)
The bases \(\Omega \approx 44.7632\) and \(\Theta \approx 215.3325\) of (3.2) serve as limiting lower bounds for root discriminants via Lemma 3.1. However it is only their square roots \(\sqrt{\Omega } \approx 6.6905 \) and \(\sqrt{\Theta } \approx 14.6742\) which Theorem 3.3 gives as limiting lower bounds for root conductors. This discrepancy will be addressed in Sect. 10.

4 Type-based analytic lower bounds

In this section we establish Theorem 4.2, which is a family of lower bounds on the root conductor \(\delta _\mathcal {X}\) of a given Artin character, dependent on the choice of an auxiliary character \(\phi \).

4.1 Conductor relations

Let G be a finite group, c an involution in G, \(\chi \) a faithful character of G, and \(\phi \) a non-zero real-valued character of G. Say that a pair of Artin characters (\(\mathcal {X}\),\(\Phi \)) has joint type \((G,c,\chi ,\phi )\) if there is a surjection \(h : {\mathbb {G}}\rightarrow G\) with \(h(\sigma )=c\), \(\mathcal {X}= \chi \circ h\), and \(\Phi = \phi \circ h\).

Write the conductors respectively as
$$\begin{aligned} D_\mathcal {X}&= \prod _p p^{c_p(\mathcal {X})}, \quad D_\Phi = \prod _p p^{c_p(\Phi )}. \end{aligned}$$
Just as in the last section, we need lower bounds on \(D_\mathcal {X}\) in terms of \(D_{\Phi }\). Our paper [11] produces bounds of this sort in the context of many characters. Here we present some of these results restricted to the setting of two characters, but otherwise following the notation of [11].
For \(\tau \in G\), let \(\bar{\tau }\) be its order. Let \(\psi \) be a rational character of G. Define two associated quantities,
$$\begin{aligned} \widehat{c}_\tau (\psi )&= \psi (e)-\psi (\tau ), \quad c_\tau (\psi ) = \psi (e) - \frac{1}{\bar{\tau }} \sum _{k|\bar{\tau }} \varphi (\bar{\tau }/k) \psi (\tau ^k). \end{aligned}$$
(4.1)
Here \(\varphi \) is the Euler totient function. For the identity element e, one clearly has \(\widehat{c}_e(\psi ) = c_e(\psi )=0\). When \(\bar{\tau }\) is prime, the functions on rational characters defined in (4.1) are proportional: \((\bar{\tau }-1) \widehat{c}_\tau (\psi ) =\bar{\tau }{c}_\tau (\psi )\).
The functions \(\widehat{c}_\tau \) and \({c}_\tau \) are related to ramification as follows. Let \(\Psi \) be an Artin character corresponding to \(\psi \) under h. If \(\Psi \) is tame at p then
$$\begin{aligned} c_p(\Psi ) = c_\tau (\psi ), \end{aligned}$$
(4.2)
for \(\tau \) corresponding to a generator of tame inertia. The identity (4.2) holds because \(c_\tau (\psi )\) is the number of non-unital eigenvalues of \(\rho (\tau )\) for a representation \(\rho \) with character \(\psi \). For general \(\Psi \), there is a canonical expansion
$$\begin{aligned} c_p(\Psi ) = \sum _{\tau } k_\tau \widehat{c}_\tau (\psi ), \end{aligned}$$
(4.3)
with always \(k_\tau \ge 0\), coming from the filtration by higher ramification groups on the p-adic inertial subgroup of G.

Because (4.1)–(4.3) are only correct for \(\psi \) rational, when we apply them to characters \(\chi \) and \(\phi \) of interest, we are always assuming that \(\chi \) and \(\phi \) are rational. As explained in Sect. 2.8, the restriction to rational characters still allows obtaining general lower bounds. Also, as will be illustrated by an example in Sect. 5.5, focusing on rational characters does not reduce the quality of these lower bounds.

For the lower bounds we need, we define the parallel quantities
$$\begin{aligned} \widehat{\alpha }(G,\chi ,\phi )&= \min _{\tau \in G-\{e\}} \frac{\widehat{c}_\tau (\chi )}{\widehat{c}_\tau (\phi )}, \quad \alpha (G,\chi ,\phi ) = \min _{\tau \in G-\{e\}} \frac{c_\tau (\chi )}{c_\tau (\phi )}. \end{aligned}$$
(4.4)
Let \(B(G,\chi ,\phi )\) be the best lower bound, valid for all primes p, that one can make on \(c_p(\mathcal {X})/c_p(\Phi )\) by purely local arguments. As emphasized in [11, §2], \(B(G,\chi ,\phi )\) can in principle be calculated by individually inspecting all possible p-adic ramification behaviors. The above discussion says
$$\begin{aligned} \widehat{\alpha }(G,\chi ,\phi ) \le B(G,\chi ,\phi ) \le \alpha (G,\chi ,\phi ). \end{aligned}$$
(4.5)
The left inequality holds because of the nonnegativity of the \(k_\tau \) in (4.3). The right inequality holds because of (4.2).
A central theme of [11] is that one is often but not always in the extreme situation
$$\begin{aligned} B(G,\chi ,\phi ) = \alpha (G,\chi ,\phi ). \end{aligned}$$
(4.6)
For example, it often occurs in practice that the minimum in the expression (4.4) for \(\widehat{\alpha }(G,\chi ,\phi )\) occurs at a \(\tau \) of prime order. Then the proportionality remark above shows that in fact all three quantities in (4.5) are the same, and so in particular (4.6) holds. As a quite different example, Theorem 7.3 of [11] says that if \(\phi \) is the regular character of G and \(\chi \) is a permutation character, then (4.6) holds. Some other examples of (4.6) are worked out in [11] by explicit analysis of wild ramification; a few examples show that \(B(G,\chi ,\phi ) < \alpha (G,\chi ,\phi )\) is possible too.

4.2 Root conductor relations

To switch the focus from conductors to root conductors, we multiply all three quantities in (4.5) by \(\phi (e)/\chi (e)\) to obtain
$$\begin{aligned} \widehat{\underline{\alpha }}(G,\chi ,\phi ) \le b(G,\chi ,\phi ) \le \underline{\alpha }(G,\chi ,\phi ). \end{aligned}$$
(4.7)
Here the elementary purely group-theoretic quantity \(\widehat{\underline{\alpha }}(G,\chi ,\phi )\) is improved to the best bound \(b(G,\chi ,\phi )\) which in turn often agrees with a second more complicated but still purely group-theoretic quantity \(\underline{\alpha }(G,\chi ,\phi )\). The notations \(\widehat{\alpha }\), \(\alpha \), \(\widehat{\underline{\alpha }}\), \(\underline{\alpha }\) are all taken from Section 7 of [11] while the notations B and b correspond to quantities not named in [11].

Our discussion establishes the following lemma.

Lemma 4.1

The conductor relation
$$\begin{aligned} \delta _\mathcal {X}\ge \delta _\Phi ^{b(G,\chi ,\phi )} \end{aligned}$$
(4.8)
holds for all pairs of Artin characters \((\mathcal {X},\Phi )\) with joint type of the form \((G,c,\chi ,\phi )\).

4.3 Bounds via an auxiliary Artin character \(\Phi \)

For \(u \in \{\widehat{\underline{\alpha }},b,\underline{\alpha }\}\), define
$$\begin{aligned} m(G,c,\chi ,\phi ,u) = M(\phi (e),\phi (c),(\phi ,1))^{u(G,\chi ,\phi )}. \end{aligned}$$
(4.9)
Just like Lemma 3.1 combined with Lemma 3.2 to give Theorem 3.3, so too Lemma 3.1 combines with Lemma 4.1 to give the following theorem.

Theorem 4.2

The lower bound
$$\begin{aligned} \delta _\mathcal {X}\ge m(G,c,\chi ,\phi ,b) \end{aligned}$$
(4.10)
is valid for all character pairs \((\mathcal {X},\Phi )\) of joint type \((G,c,\chi ,\phi )\) such that \(\Phi \) is non-negative and \(L(\Phi ,s)\) satisfies the Artin conjecture and the Riemann hypothesis.

Computing the right side of (4.10) is difficult because the base in (4.9) requires evaluating the maximum of a complicated function, while the exponent \(b(G,\chi ,\phi )\) involves an exhaustive study of wild ramification. Almost always in the sequel, \(\chi \) and \(\phi \) are rational-valued and we replace \(b(G,\chi ,\phi )\) by \(\widehat{\underline{\alpha }}(G,\chi ,\phi )\); in the common case that all three quantities of (4.7) are equal, this is no loss.

5 Four choices for \(\phi \)

This section fixes a type \((G,c,\chi )\) where the faithful character \(\chi \) is rational-valued and uses the notation \((n,r) = (\chi (e),\chi (c))\). The section introduces four nonnegative characters \(\phi _i\) built from \((G,\chi )\). For the first character \(\phi _L\), it makes \(m(G,c,\chi ,\phi _L,b)\), the lower bound appearing in Theorem 4.2, more explicit. For the remaining three characters \(\phi _i\), it makes the perhaps smaller quantity \(m(G,c,\chi ,\phi _i,\widehat{\underline{\alpha }})\) more explicit.

Two simple quantities enter into the constructions as follows. Let X be the set of values of \(\chi \), so that \(X \subset \mathbf {Z}\) by our rationality assumption. Let be the least element of X. The greatest element of X is of course \(\chi (e)=n\), and we let \(\widehat{\chi }\) be the second greatest element. Thus, .

5.1 Linear auxiliary character

A simple nonnegative character associated to \(\chi \) is . Both \(m(G,c,\chi ,\phi _L,\widehat{\underline{\alpha }})\) and \(m(G,c,\chi ,\phi _L,\underline{\alpha })\) easily evaluate to
(5.1)
The character \(\phi _L\) seems most promising as an auxiliary character when is very small.

In [24, §3] the auxiliary character \(\chi +n\) is used, which has the advantage of being nonnegative for any rational character \(\chi \). Odlyzko also uses \(\chi +n\) in [20], and suggests using the auxiliary character since it gives a better bound whenever . This strict inequality holds exactly when the center of G has odd order.

5.2 Square auxiliary character

Another very simple nonnegative character associated to \(\chi \) is \(\phi _S = \chi ^2\). This character gives
$$\begin{aligned} m(G,c,\chi ,\phi _S,\widehat{\underline{\alpha }}) = M(n^2,r^2,(\chi ,\chi ))^{n/(n+\widehat{\chi })}. \end{aligned}$$
(5.2)
The derivation of (5.2) uses the simple formula in (4.1) for \(\widehat{c}_\tau \). The formula for \(c_\tau \) in (4.1) is more complicated, and we do not expect a simple general formula for \(m(G,c,\chi ,\phi _S,{\underline{\alpha }})\), nor for the best bound \(m(G,c,\chi ,\phi _S,b)\) in Theorem 4.2.

The character \(\phi _S\) is used prominently in [20, 24]. When \(\widehat{\chi }=n-2\), the lower bound \(m(G,c,\chi ,\phi _S,\widehat{\underline{\alpha }})\) coincides with that of Lemma 3.2. Thus for \(\widehat{\chi }=n-2\), Theorem 4.2 with \(\phi = \phi _S\) gives the same bound as Theorem 3.3. On the other hand, as soon as \(\widehat{\chi }<n-2\), Theorem 4.2 with \(\phi = \phi _S\) is stronger. The remaining case \(\widehat{\chi }=n-1\) occurs only three times among the 195 characters we consider in Sect. 8. In these three cases, the bound in Theorem 3.3 is stronger because the exponent is larger. However, in each of these cases, the tame-wild principle applies [11] and we can use exponent \(m(G,c,\chi ,\phi _S,{\underline{\alpha }})\), which gives the same bound as Theorem 3.3 in two cases, and a better bound in the third.

5.3 Quadratic auxiliary character

Let \(-\widetilde{\chi }\) be the greatest negative element of the set X of values of \(\chi \). A modification of the given character \(\chi \) is \(\chi ^* = \chi +\widetilde{\chi }\), with degree \(n^* = n+\widetilde{\chi }\) and signature \(r^* = r + \widetilde{\chi }\). A modification of \(\phi _S\) is \(\phi _Q = \chi \chi ^*\). The function \(\phi _Q\) takes only nonnegative values because the interval \((-\tilde{\chi },0)\) in the x-line where \(x(x+\tilde{\chi })\) is negative is disjoint from the set X of values of \(\chi \). The lower bound associated to \(\phi _Q\) is
$$\begin{aligned} m(G,c,\chi ,\phi _Q,\widehat{\underline{\alpha }}) = M(nn^*,rr^*,(\chi ,\chi ))^{(n^*)/(n^*+\widehat{\chi })}. \end{aligned}$$
(5.3)
Comparing formulas (5.2) and (5.3), \(n^2\) strictly increases to \(nn^*\) and \(n/(n+\widehat{\chi })\) increases to \(n^*/(n^* + \widehat{\chi })\). In the totally real case \(n=r\), the monotonicity of the function M(nn) as exhibited in the right edge of Fig. 1 then implies that \(m(G,c,\chi ,\phi _S,\widehat{\underline{\alpha }})\) strictly increases to \(m(G,c,\chi ,\phi _Q,\widehat{\underline{\alpha }})\). Even outside the totally real setting, one can expect that \(\phi _Q\) almost always yields a better lower bound than \(\phi _S\). The character \(\phi _Q\) seems promising as an auxiliary character when \(\widehat{\chi }\) is very small so that the exponent is near 1 rather than its lower limit of 1 / 2. As for the square case, we do not expect a simple formula for the best bound \(m(G,c,\chi ,\phi _Q,b)\) in Theorem 4.2.

5.4 Galois auxiliary character

Finally there is a strong candidate for a good auxiliary character that does not depend on \(\chi \), namely the regular character \(\phi _G\). By definition, \(\phi _G(e) = |G|\) and else \(\phi _G(g)=0\). In this case one has
$$\begin{aligned} m(G,c,\chi ,\phi _G,\widehat{\underline{\alpha }}) = M(|G|,\delta _{ce}|G|,1)^{(n-\widehat{\chi })/n}. \end{aligned}$$
(5.4)
Here \(\delta _{ce}\) is defined to be 1 in the totally real case and 0 otherwise. This auxiliary character again seems most promising when \(\widehat{\chi }\) is small. As in the square and quadratic cases, we do not expect a simple formula for \(m(G,c,\chi ,\phi _G,b)\).

5.5 Spectral bounds and rationality

To get large lower bounds on root conductors, one wants to be small for (5.1) or \(\widehat{\chi }/n\) to be small for (5.2)–(5.4). The analogous quantities and \(\widehat{\chi }/n_1\) are well-defined for a general real character \(\chi _1\), and replacing \(\chi _1\) by the sum \(\chi \) of its conjugates can substantially reduce them.

For example, let p be a prime congruent to 1 modulo 4, and let G be the simple group \(\mathrm {PSL}_2(p)\). Then G has two irrational irreducible characters, say \(\chi _1\) and \(\chi _2\), both of degree \((p+1)/2\). For each, its set of values is
$$\begin{aligned} \left\{ \frac{-\sqrt{p}-1}{2},-1,0,1,\frac{\sqrt{p}-1}{2}, \frac{p+1}{2}\right\} \end{aligned}$$
(except that 1 is missing if \(p=5\)). However for \(\chi = \chi _1+\chi _2\), the set of values is just \(\{-2,0,2,p+1\}\). Thus in passing from to , one saves a factor of \(\sqrt{p}+1\). Similarly in passing from \(\widehat{\chi }_1/n_1\) to \(\widehat{\chi }/n\), one saves a factor of \(\sqrt{p}-1\).

6 Other choices for \(\phi \)

To apply Theorem 4.2 for a given Galois type \((G,c,\chi )\), one needs to choose an auxiliary character \(\phi \). We presented four choices in Sect. 5. We discuss all possible choices here, using \(G=A_4\) and \(G=A_5\) as illustrative examples.

6.1 Rational character tables

As a preliminary, we review the notion of rational character table. Let \(G^\sharp = \{C_j\}_{j \in J}\) be the set of power-conjugacy classes in G. Let \(G^\mathrm{rat} = \{\chi _i\}_{i \in I}\) be the set of rationally irreducible characters. These sets have the same size k and one has a \(k\times k\) matrix \(\chi _i(C_j)\), called the rational character table.
Table 1

Rational character tables for \(A_4\) and \(A_5\)

Two examples are given in Table 1. We index characters by their degree, with \(I = \{1,2,3\}\) for \(A_4\) and \(I = \{1,4,5,6\}\) for \(A_5\). All characters are absolutely irreducible except for \(\chi _2\) and \(\chi _{6}\), which each break as a sum of two conjugate irreducible complex characters. We likewise index power-conjugacy classes by the order of a representing element, always adding letters as is traditional. Thus \(J = \{1A,2A,3AB\}\) for \(A_4\) and \(J = \{1A,2A,3A,5AB\}\) for \(A_5\), with 3AB and 5AB each consisting of two conjugacy classes.

6.2 The polytope \(P_G\) of normalized nonnegative characters

A general real-valued function \(\phi \in \mathbf {R}(G^\sharp )\) has an expansion \(\sum x_i \chi _i\) with \(x_i \in \mathbf {R}\). The coefficients are recovered via inner products, \(x_i = (\phi ,\chi _i)/(\chi _i,\chi _i)\). Alternative coordinates are given by \(y_j = \phi (C_j)\). The \(\phi \) allowed for Theorem 4.2 are the nonzero \(\phi \) with the \(x_i\) and \(y_j\) non-negative integers.

An allowed \(\phi \) gives the same lower bound in Theorem 4.2 as any of its positive multiples \(m \phi \). Without getting any new bounds, we can therefore give ourselves the convenience of allowing the \(x_i\) and \(y_j\) to be nonnegative rational numbers. Similarly, we can extend by continuity to allow the \(x_i\) and \(y_j\) to be nonnegative real numbers. The allowed \(\phi \) then become the cone in k-dimensional Euclidean space given by \(x_i \ge 0\) and \(y_j \ge 0\), excluding the tip of the cone at the origin.

Writing the identity character as \(\chi _1\), we can normalize via scaling to \(x_1=1\). Writing the identity class as \(C_{1A}\), the inequality \(y_{1A} \ge 0\) is implied by the other \(y_j \ge 0\) and so the variable \(y_{1A}\) can be ignored. The polytope \(P_G\) of normalized nonnegative characters is then defined by \(x_1=1\), the inequalities \(x_i \ge 0\) for \(i \ne 1\), and inequalities \(y_j \ge 0\) for \(j \ne 1A\). The point where all the \(x_i\) are zero is the unital character \(\phi _1\). The point where all the \(y_j\) are zero is the regular character \(\phi _G\). Thus the \((k-1)\)-dimensional polytope \(P_G\) is determined by \(2k-2\) linear inequalities, with \(k-1\) corresponding to non-unital characters and intersecting at \(\phi _1\), and \(k-1\) corresponding to non-identity classes and intersecting at \(\phi _G\).
Fig. 2

The polytopes \(P_{A_4}\) and \(P_{A_5}\)

Figure 2 continues our two examples. On the left, \(P_{A_4}\) is drawn in the \(x_2\)\(x_3\) plane. The character faces give the coordinate axes and are dashed. The class faces are calculated from columns in the rational character table and are solid. On the right, a view of \(P_{A_5}\) is given in \(x_{4}\)\(x_{5}\)\(x_{6}\) space. The three pairwise intersections of character faces give coordinate axes and are dashed, while all other edges are solid. In this view, the point \(\phi _G = \phi _{60} = (4,5,3)\) should be considered as closest to the reader, with the solid lines visible and the dashed lines hidden by the polytope. Note that \(P_{A_4}\) has the combinatorics of a square and \(P_{A_5}\) has the combinatorics of a cube. While the general \(P_G\) is the intersection of an orthant with tip \(\phi _1\) and an orthant with tip \(\phi _G\), its combinatorics are typically more complicated than \([0,1]^{(k-1)}\). For example, the groups \(G=A_6\), \(S_5\), \(A_7\), and \(S_6\), have \(k=6\), 7, 8, and 11 respectively; but instead of having 32, 64, 128 and 1024 vertices, their polytopes \(P_G\) have 28, 40, 115, and 596 vertices respectively.

6.3 Points in \(P_G\)

In the previous subsection, we have mentioned already the distinguished vertices \(\phi _1\) and \(\phi _G\). For every rationally irreducible character, we also have , \(\phi _{\chi ,S} = \chi ^2\), and \(\phi _{\chi ,Q} = \chi \chi ^*\), as in Sect. 5.

For every subgroup H of G, another element of \(P_G\) is the permutation character \(\phi _{G/H}\). For \(H = G\), this character is just the \(\phi _1\) considered before, which is a vertex. Otherwise, a theorem of Jordan, discussed at length in [30], says that \(\phi _{G/H}(C_j)=0\) for at least one j; in other words, \(\phi _{G/H}\) is on at least one character face. For \(A_4\) and \(A_5\), there are respectively five and nine conjugacy classes of subgroups, distinguished by their orders. Figure 2 draws the corresponding points, labeled by \(\phi _{|G/H|}\). All four vertices of \(P_{A_4}\) and six of the eight vertices of \(P_{A_5}\) are of the form \(\phi _{N}\). The remaining one \(\phi _N\) in \(P_{A_4}\) is on an edge, while the remaining three \(\phi _N\) in \(P_{A_5}\) are on edges as well.

6.4 The best choice for \(\phi \)

Given \((G,c,\chi )\) and \(u \in \{\widehat{\underline{\alpha }},b,\underline{\alpha }\}\), let \(m(G,c,\chi ,u) = \max _{\phi \in P_G} m(G,c,\chi ,\phi ,u)\). Computing these maxima seems difficult. Instead we vary \(\phi \) over a modestly large finite set, denoting the largest bound appearing as \(\mathfrak {d}(G,c,\chi ,u)\). For most G, the set of \(\phi \) we inspect consists of all \(\phi _{\chi ,L}\), \(\phi _{\chi ,S}\), and \(\phi _{\chi ,Q}\), all \(\phi _{G/H}\) including the regular character \(\phi _G\), and all vertices. For some G, like \(S_7\), there are too many vertices and we exclude them from the list of \(\phi \) we try.

For each \((G,\chi )\), we work either with \(u=\widehat{\underline{\alpha }}\) or with \(u=\underline{\alpha }\), as explained in the “middle four columns” part of Sect. 8.2.2. We then report \(\mathfrak {d}(G,\chi ) = \min _c \mathfrak {d}(G,c,\chi ,u)\) in Sect. 8.

7 The case \(G=S_5\)

Our focus in the next two sections is on finding initial segments \(\mathcal {L}(G,\chi ; B)\) of complete lists of Artin L-functions, and in particular on finding the first root conductor \(\delta _1(G,\chi )\). It is a question of transferring completeness statements for number fields to completeness statements for Artin L-functions via conductor relations. In this section, we explain the process by presenting the case \(G=S_5\) in some detail.

7.1 Different orders on the same set of fields

Consider the set \(\mathcal {K}\) of isomorphism classes of quintic fields K over \(\mathbf {Q}\) with splitting field \(L/\mathbf {Q}\) having Galois group \({{\mathrm{Gal}}}(L/\mathbf {Q}) \cong S_5\). The group \(S_5\) has seven irreducible characters which we index by degree and an auxiliary label: \(\chi _{1a} = 1\), \(\chi _{1b}\), \(\chi _{4a}\), \(\chi _{4b}\), \(\chi _{5a}\), \(\chi _{5b}\), and \(\chi _{6a}\). For \(\phi \) a permutation character, let \(D_\phi (K) = D(K_\phi )\) be the absolute discriminant of the associated resolvent algebra \(K_\phi \) of K. Extending by multiplicativity, functions \(D_\chi : \mathcal {K}\rightarrow \mathbf {R}_{>0}\) are defined for general \(\chi = \sum m_n \chi _n\). They do not depend on the coefficient \(m_{1a}\). We follow our practice of often shifting attention to the corresponding root conductors \(\delta _\chi (K) = D_\chi (K)^{1/\chi (e)}\).

Let \(\mathcal {K}(\chi ; B) = \{K \in \mathcal {K}: \delta _\chi (K) \le B\}\). Suppose now all the \(m_{n}\) are nonnegative with at least one coefficient besides \(m_{1a}\) and \(m_{1b}\) positive. Then \(\delta _\chi \) is a height function in the sense that all the \(\mathcal {K}(\chi ; B)\) are finite. Suppressing the secondary phenomenon that ties among a finite number of fields can occur, we think of each \(\delta _\chi \) as giving an ordering on the set \(\mathcal {K}\).

The orderings coming from different \(\delta _\chi \) can be very different. For example, consider the field \(K \in \mathcal {K}\) defined by the polynomial \(x^5 - 2x^4 + 4x^3 - 4x^2 + 2x - 4\). This field is the first field in \(\mathcal {K}\) when ordered by the regular character \(\phi _{120} = \sum _n \chi _n(n) \chi _n\). However it is the 22nd field when ordered by \(\phi _6 = 1 + \chi _{5b}\) only the 2298th field when ordered by \(\phi _5 = 1+ \chi _{4a}\).

This phenomenon of different orderings on the same set of number fields plays a prominent role in asymptotic studies [32]. Here we are interested instead in initial segments and how they depend on \(\chi \). Our formalism lets us treat any \(\chi \). Following the conventions for general G of the next section, we focus on the five irreducible \(\chi \) with \(\chi (e)>1\), thus \(\chi _n\) for \(n \in \{4a,4b,5a,5b,6a\}\).

7.2 Computing Artin conductors

To compute general \(D_\chi (K)\), one needs to work with enough resolvents of \(K=K_5 = \mathbf {Q}[x]/f_5(x)\). For starters, we have the quadratic resolvent \(K_2 = \mathbf {Q}[x]/(x^2-D(K_5))\) and the Cayley-Weber resolvent \(K_6 = \mathbf {Q}[x]/f_6(x)\) [8, 11]. The other resolvents we will need are \(K_{10} = K_5 \otimes K_2\), \(K_{12} = K_2 \otimes K_6\), and \(K_{30} = K_5 \otimes K_6\). Defining polynomials are obtained for \(K_a \otimes K_b\) by the general formula
$$\begin{aligned} f_{ab}(x) = \prod _{i=1}^a \prod _{j=1}^b (x-\alpha _i-\beta _j), \end{aligned}$$
where \(f_a(x)\) has roots \(\alpha _i\) and \(f_b(x)\) has roots \(\beta _j\). So discriminants \(D_2\), \(D_5\), \(D_6\), \(D_{10}\), \(D_{12}\), \(D_{30}\) are easily computed.
From the character table, the permutation characters \(\phi _N\) in question are expressed in the basis \(\chi _n\) as on the left in the following display. Inverting, one gets the \(\chi _n\) in terms of the \(\phi _N\) as on the right.
$$\begin{aligned} \begin{array}{lllll} \phi _2 &{}&{} = 1 + \chi _{1b}, &{}\chi _{1b} &{} = -1 + \phi _2,\\ \phi _5 &{}&{} = 1 + \chi _{4a}, &{} \chi _{4a} &{} = -1 + \phi _{5}, \\ \phi _6 &{}&{} = 1 + \chi _{5b}, &{} \chi _{4a} &{} = 1 - \phi _2 - \phi _{5} + \phi _{10}, \\ \phi _{10} &{} = \phi _{5} \phi _2 &{} = 1 + \chi _{1b} + \chi _{4a} + \chi _{4b}, &{} \chi _{5a} &{} = 1 - \phi _2 - \phi _6 + \phi _{12}, \\ \phi _{12} &{} = \phi _6 \phi _2 &{} = 1 + \chi _{1b} + \chi _{5a} + \chi _{5b}, &{} \chi _{5b} &{} = -1 + \phi _6, \\ \phi _{30} &{} = \phi _5 \phi _6 &{} = 1 + 2 \chi _{4a} + 2 \chi _{5a} + \chi _{5b} + \chi _{6a}, \quad &{} \chi _{6a} &{} = 2 \phi _2 - 2 \phi _5 + \phi _6 - 2 \phi _{12} + \phi _{30}. \end{array} \end{aligned}$$
Conductors \(D_n\) belonging to the \(\chi _n\) are calculable through these formulas, as e.g. \(D_{6a} = D_2^2 D_5^{-2} D_6 D_{12}^{-2} D_{30}\).

For all the groups G considered in the next section, we proceeded similarly. Thus we started with rational character tables from Magma. We used linear algebra to express rationally irreducible characters in terms of permutation characters. We used Magma again to compute resolvents and then Pari to evaluate their discriminants. In this last step, we often confronted large degree polynomials with large coefficients. The discriminant computation was only feasible because we knew a priori the set of primes dividing the discriminant, and could then easily compute the p-parts of the discriminants of these resolvent fields for relevant primes p using Pari/gp without fully factoring the discriminants of the resolvent polynomials.

Magma’s Artin representation package computes conductors of Artin representations in a different and more local manner. Presently, it does not compute all conductors in our range because some decomposition groups are too large.

7.3 Transferring completeness results

As an initial complete list of fields, we take \(\mathcal {K}(\phi ; 85)\) with \(\phi =\phi _G=\phi _{120}\). We know from [10] that this set consists of 2080 fields. We list these fields by increasing discriminant, \(K^1\), ..., \(K^{2080}\), with the resolution of ties conveniently not affecting the explicit results appearing in Table 3.
Table 2

Standard character table of \(S_5\) on the left, with entries \(\chi _n(\tau )\); tame table [11, §4.3], on the right, with entries \(c_\tau (\chi _n)\) as defined in (4.1)

The quantities of Sect. 4 reappear here, and we will use the abbreviations \(\widehat{\underline{\alpha }}(n) = \widehat{\underline{\alpha }}(S_5,\chi _n,\phi )\) and \(\underline{\alpha }(n) = \underline{\alpha }(S_5,\chi _n,\phi )\). Since \(\phi \) is zero outside of the identity class, the formulas simplify substantially:
$$\begin{aligned} \widehat{\underline{\alpha }}(n)&=\frac{\phi (e)}{\chi _n(e)} \min _\tau \frac{\chi _n(e)-\chi _n(\tau )}{\phi (e)-\phi (\tau )} = 1-\max _{\tau } \frac{\chi _n(\tau )}{n}, \\ \underline{\alpha }(n)&= \frac{\phi (e)}{\chi _n(e)} \min _{\tau } \frac{c_\tau (\chi _{n})}{c_\tau (\phi )} = \frac{1}{n} \min _{\tau } \frac{c_\tau (\chi _{n}) \overline{\tau }}{\overline{\tau } - 1}. \end{aligned}$$
For each of the five n, the classes contributing to the minima are in bold on Table 2. So, extremely simply, for computing \(\widehat{\underline{\alpha }}(n)\) on the left, the largest \(\chi _n(\tau )\) besides \(\chi _n(e)\) are in italic. For computing \(\underline{\alpha }(n)\) on the right, the \(c_\tau (\chi _n)\) with \(c_\tau (\chi _n)/c_\tau (\phi )\) minimized are put in bold. For the group \(S_5\), one has agreement \(\widehat{\underline{\alpha }}(n) = \underline{\alpha }(n)\) in all five cases. This equality occurs for 170 of the lines in Tables 3, 4, 5, 6 and 7, with the other possibility \(\widehat{\underline{\alpha }}(n) < \underline{\alpha }(n)\) occurring for the remaining 25 lines.
For any cutoff B, conductor relations give
$$\begin{aligned} \mathcal {K}(\chi _n;B^{\widehat{\underline{\alpha }}(n)}) \subseteq \mathcal {K}(\phi ; B). \end{aligned}$$
One has an analogous inclusion for general \((G,\chi )\), with \(\phi \) again the regular character for G. When G satisfies the tame-wild principle of [11], the \(\widehat{\underline{\alpha }}\) in exponents can be replaced by \(\underline{\alpha }\). The group \(S_5\) does satisfy the tame-wild principle, but in this case the replacement has no effect.

The final results are on Table 3. In particular for \(n = 4a\), 4b, 5a, 5b, 6a the unique minimizing fields are \(K^{103}\), \(K^{21}\), \(K^{14}\), \(K^{6}\), and \(K^{12}\), with root conductors approximately 6.33, 18.72, 17.78, 16.27, and 18.18. The lengths of the initial segments identified are 45, 15, 186, 592, and 110. Note that because of the relations \(\phi _5 = 1+ \chi _{4a}\) and \(\phi _6 = 1 + \chi _{5b}\), the results for 4a and 5b are just translation of known minima of discriminants of number fields with Galois groups 5T5 and 6T14 respectively. For 4b, 5b, 6a, and the majority of the characters in the tables of the next section, the first root conductor and the entire initial segment are new.

8 Tables for 84 groups G

In this section, we present our computational results for small Galois types. For simplicity, we focus on results coming from complete lists of Galois number fields. Summarizing statements are given in Sect. 8.1 and then many more details in Sect. 8.2.

8.1 Lower bounds and initial segments

We consider all groups with a faithful transitive permutation representation in some degree from two to nine, except we exclude the nonsolvable groups in degrees eight and nine. There are 84 such groups, and we consider all associated Galois types \((G,\chi )\) with \(\chi \) a rationally irreducible faithful character. Our first result gives conditional lower bounds:

Theorem 8.1

For each of the 195 Galois types \((G,\chi )\) listed in Tables 3, 4, 5, 6 and 7, the listed value \(\mathfrak {d}\) gives a lower bound for the root conductor of all Artin representations of type \((G,\chi )\), assuming the Artin conjecture and Riemann hypothesis for relevant L-functions.

The bounds in Tables 3, 4, 5, 6 and 7 are graphed with the best previously known bounds from [24] in Fig. 3. The horizontal axis represents the dimension \(n_1=\chi _1(e)\) of any irreducible constituent \(\chi _1\) of \(\chi \). The vertical axis corresponds to lower bounds on root conductors. The piecewise-linear curve connects bounds from [24], and there is one dot at height \(\mathfrak {d}(G,\chi )\) for each \((G,\chi )\) from Tables 3, 4, 5, 6 and 7 with \(\chi _1(e)\le 20\). Here we are freely passing back and forth between a rational character \(\chi \) and an irreducible constituent \(\chi _1\) via \(\delta _1(G,\chi ) = \delta _1(G,\chi _1)\), which is a direct consequence of (2.4).

Not surprisingly, the type-based bounds are larger. In low dimensions \(n_1\), some type-based bounds are close to the general bounds, but by dimension 5 there is a clear separation which widens as the dimension grows. This may in part be explained by the fact that we are only seeing a small number of representations for each of these dimensions. However, as we explain in Sect. 10.3, we also expect that the asymptotic lower bound of \(\sqrt{\Omega } \approx 6.7\) [24] is not optimal, and that this bound is more likely to be at least \(\Omega \approx 44.8\).
Fig. 3

Points \((\chi _1(e),\mathfrak {d}(G,\chi ))\) present lower bounds from Tables 3, 4, 5, 6and 7. The piecewise-linear curve plots lower bounds from [24]. Both the points and the curve assume the Artin conjecture and Riemann hypothesis for the relevant L-functions

Table 3

Artin L-functions with small conductor from groups in degrees 2, 3, 4, and 5

\(\varvec{G}\)

\(\varvec{n_1}\)

\(\varvec{z}\)

\(\varvec{\mathfrak {d}}\)

\(\varvec{\delta _1}\)

\(\varvec{\Delta _1}\)

Pos’n

\(\varvec{\beta }\)

\(\varvec{B^\beta }\)

\(\varvec{\#}\)

\(C_2\)

2

TW

2, 2

 

1.73

\(3^*\)

  

100

6086

2T1

1

 

\([-1, -1]\)

\(2.97_{\ell }\)

3.00

3

1

2.00

10,000

6086

\(C_3\)

3

TW

2, 2

 

3.66

\(7^*\)

  

500

1772

3T1

1

\(\sqrt{-3}\)

\([-1, -1]\)

\(6.93_{\ell }\)

7.00

7

1

1.50

11180.34

1772

\(S_3\)

6

TW

3, 4

 

4.80

\(23^*\)

  

250

24,484

3T2

2

 

\([-1, 0]\)

\(4.74_{\ell }\)

4.80

23

1

1.00

250

13,329

\(C_4\)

4

TW

3, 4

 

3.34

\(5^*\)

  

150

2668

4T1

1

i

\([-2, 0]\)

\(4.96_{S}\)

5.00

5

1

\(1.33_\bullet \)

796.99

489

\(D_4\)

8

TW

5, 10

 

6.03

\(3^*7^*\)

  

150

31,742

4T3

2

 

\([-2, 0]\)

\(5.74_{q}\)

6.24

\(3 \cdot 13\)

2

1.00

150

9868

\(A_4\)

12

TW

3, 4

 

10.35

\(2^*7^*\)

  

150

846

4T4

3

 

\([-1, 0]\)

\(7.60_{q}\)

14.64

\(2^{6} 7^{2}\)

1

1.00

150

270

\(S_4\)

24

TW

5, 12

 

13.56

\(2^*11^*\)

  

150

14,587

6T8

3

 

\([-1, 1]\)

\(8.62_{G}\)

11.30

\(2^{2} 19^{2}\)

4

\(0.89_\bullet \)

85.96

779

4T5

3

 

\([-1, 1]\)

\(5.49_{p}\)

6.12

229

9

0.67

28.23

1603

\(C_5\)

5

TW

2, 2

 

6.81

\(11^*\)

  

200

49

5T1

1

\(\zeta _{5}\)

\([-1, -1]\)

\(10.67_{\ell }\)

11.00

11

1

1.25

752.12

49

\(D_5\)

10

TW

3, 4

 

6.86

\(47^*\)

  

200

3622

5T2

2

\(\sqrt{5}\)

\([-1, 0]\)

\(6.73_{q}\)

6.86

47

1

1.00

200

3219

\(F_5\)

20

TW

4, 8

 

11.08

\(2^*5^*\)

  

200

3010

5T3

4

 

\([-1, 0]\)

\(10.28_{q}\)

13.69

\(2^{4} 13^{3}\)

2

1.00

200

2066

\(A_5\)

60

TW

4, 8

 

18.70

\(2^*17^*\)

  

85

473

5T4

4

 

\([-1, 1]\)

\(8.18_{g}\)

11.66

\(2^{6} 17^{2}\)

1

0.75

27.99

46

6T12

5

 

\([-1, 1]\)

\(10.18_{p}\)

12.35

\(2^{6} 67^{2}\)

3

0.80

34.96

216

12T33

3

\(\sqrt{5}\)

\([-2, 1]\)

\(10.34_{g}\)

26.45

\(2^{6} 17^{2}\)

1

0.83

40.54

18

\(S_5\)

120

TW

7, 40

 

24.18

\(2^*3^*5^*\)

  

85

2080

5T5

4

 

\([-1, 2]\)

\(6.28_{\ell }\)

6.33

1609

103

0.50

9.22

45

10T12

4

 

\([-2, 1]\)

\(10.28_{V}\)

18.72

\(5^{2} 17^{3}\)

21

0.75

27.99

15

10T13

5

 

\([-1, 1]\)

\(12.13_{V}\)

16.27

\(2^{4} 3^{2} 89^{2}\)

6

0.80

34.96

592

6T14

5

 

\([-1, 1]\)

\(11.09_{g}\)

17.78

\(2^{6} 3^{4} 7^{3}\)

14

0.80

34.96

186

20T35

6

 

\([-2, 1]\)

\(12.26_{g}\)

18.18

\(2^{4} 3^{3} 17^{4}\)

12

0.83

40.54

110

Table 4

Artin L-functions of small conductor from sextic groups

\(\varvec{G}\)

\(\varvec{n_1}\)

\(\varvec{z}\)

\(\varvec{\mathfrak {d}}\)

\(\varvec{\delta _1}\)

\(\varvec{\Delta _1}\)

Pos’n

\(\varvec{\beta }\)

\(\varvec{B^\beta }\)

\(\varvec{\#}\)

\(C_6\)

6

TW

4, 6

 

5.06

\(7^*\)

  

200

9609

6T1

1

\(\sqrt{-3}\)

\([-2, 1]\)

\(6.93_{P}\)

7

7

1

\(1.20_\bullet \)

577.08

617

\(D_6\)

12

TW

6, 14

 

8.06

\(3^*5^*\)

  

150

46,197

6T3

2

 

\([-2, 1]\)

\(7.60_{G}\)

9.33

\(3 \cdot 29\)

6

\(1_\bullet \)

150

10,242

\(S_3C_3\)

18

 

6, 17

 

10.06

\(2^*3^*7^*\)

  

200

9420

6T5

2

\(\sqrt{-3}\)

\([-2, 1]\)

\(5.69_{q*}\)

7.21

\(2^{2} 13\)

4

0.75

53.18

503

\(A_4C_2\)

24

 

6, 16

 

12.31

\(2^*7^*\)

  

150

6676

6T6

3

 

\([-3, 1]\)

\(7.60_{p}\)

8.60

\(7^{2} 13\)

3

0.67

28.23

98

\(S_3^2\)

36

 

9, 69

 

15.53

\(2^*19^*\)

  

200

45,117

6T9

4

 

\([-2, 1]\)

\(7.98_{q*}\)

14.83

\(2^{4} 5^{2} 11^{2}\)

27

0.75

53.18

824

\(C_3^2{\rtimes }C_4\)

36

TW

5, 16

 

23.57

\(3^*5^*\)

  

150

331

6T10

\(4^{{{2}}}\)

 

\([-2, 1]\)

\(7.98_{q*}\)

17.80

\(2^{11} 7^{2}\)

2

0.75

42.86

33

\(S_4C_2\)

48

 

10, 96

 

16.13

\(2^*23^*\)

  

150

70,926

6T11

\(3^{{{2}}}\)

 

\([-3, 1]\)

\(6.14_{g}\)

6.92

\(2^{2} 83\)

7

0.67

28.23

3694

\(C_3^2{\rtimes }D_4\)

72

TW

9, 105

 

21.76

\(3^*11^*\)

  

150

8536

6T13

\(4^{{{2}}}\)

 

\([-2, 2]\)

\(7.60_{p}\)

7.90

\(3^{2} 433\)

52

0.50

12.25

41

12T36

\(4^{{{2}}}\)

 

\([-2, 1]\)

\(11.29_{P}\)

23.36

\(3^{5} 5^{2} 7^{2}\)

18

0.75

42.86

106

\(A_6\)

360

TW

6, 28

 

31.66

\(2^*3^*\)

  

60

26

6T15

\(5^{{{2}}}\)

 

\([-1, 2]\)

\(7.71_{\ell }\)

12.35

\(2^{6} 67^{2}\)

8

0.60

11.67

0

10T26

9

 

\([-1, 1]\)

\(17.69_{g}\)

28.20

\(2^{18} 3^{16}\)

1

0.89

38.07

7

30T88

10

 

\([-2, 1]\)

\(18.34_{g}\)

30.61

\(2^{24} 3^{16}\)

1

0.90

39.84

4

36T555

8

\(\sqrt{5}\)

\([-2, 1]\)

\(20.70_{g}\)

42.81

\(2^{18} 3^{16}\)

1

0.94

46.45

3

\(S_6\)

720

 

11, 596

 

33.50

\(2^*3^*5^*\)

  

60

99

12T183

\(5^{{{2}}}\)

 

\([-3, 2]\)

\(8.21_{v}\)

11.53

\(11^{2} 41^{2}\)

6

0.60

11.67

1

6T16

\(5^{{{2}}}\)

 

\([-1, 3]\)

\(6.23_{\ell }\)

6.82

14731

53

0.40

5.14

0

10T32

9

 

\([-1, 3]\)

\(10.77_{v}\)

16.60

\(2^{15} 11^{3} 13^{3}\)

74

0.67

15.33

0

20T145

9

 

\([-3, 1]\)

\(19.33_{g}\)

31.25

\(2^{6} 5^{6} 73^{4}\)

16

0.89

38.07

4

30T176

\(10^{{{2}}}\)

 

\([-2, 2]\)

\(16.88_{v}\)

24.22

\(11^{4} 41^{6}\)

6

0.80

26.46

1

36T1252

16

 

\([-2, 1]\)

\(22.73_{g}\)

35.46

\(2^{36} 3^{8} 7^{12}\)

5

0.94

46.45

11

Table 5

Artin L-functions of small conductor from septic groups

\(\varvec{G}\)

\(\varvec{n_1}\)

\(\varvec{z}\)

\(\varvec{\mathfrak {d}}\)

\(\varvec{\delta _1}\)

\(\varvec{\Delta _1}\)

Pos’n

\(\varvec{\beta }\)

\(\varvec{B^\beta }\)

\(\varvec{\#}\)

\(C_7\)

7

TW

2, 2

 

17.93

\(29^*\)

  

200

15

7T1

1

\(\zeta _{7}\)

\([-1, -1]\)

\(14.03_{\ell }\)

29

29

1

1.17

483.65

15

\(D_7\)

14

TW

3, 4

 

8.43

\(71^*\)

  

200

2078

7T2

2

\(\zeta _7^+\)

\([-1, 0]\)

\(8.38_{q}\)

8.43

71

1

1

200

1948

\(C_7{\rtimes }C_3\)

21

TW

3, 4

 

31.64

\(2^*73^*\)

  

100

11

7T3

3

\(\sqrt{-7}\)

\([-1, 0]\)

\(25.50_{q}\)

34.93

\(2^{3} 73^{2}\)

1

1

100

8

\(F_7\)

42

TW

5, 12

 

15.99

\(2^*7^*\)

  

75

342

7T4

6

 

\([-1, 0]\)

\(14.47_{q}\)

18.34

\(11^{3} 13^{4}\)

2

1

75

287

\(\text {GL}_3(2)\)

168

TW

5, 14

 

32.25

\(2^*3^*11^*\)

  

45

19

42T37

3

\(\sqrt{-7}\)

\([-2, 2]\)

\(15.55_{G}\)

26.06

\(7^{2} 19^{2}\)

7

\(0.89_\bullet \)

29.48

1

7T5

6

 

\([-1, 2]\)

\(9.36_{p}\)

11.23

\(13^{2} 109^{2}\)

4

0.67

12.65

1

8T37

7

 

\([-1, 1]\)

\(14.10_{g}\)

32.44

\(3^{8} 7^{8}\)

11

0.86

26.12

0

21T14

8

 

\([-1, 1]\)

\(14.90_{g}\)

23.16

\(2^{6} 3^{6} 11^{6}\)

1

0.88

27.96

1

\(A_7\)

2520

 

8, 115

 

39.52

\(2^*3^*7^*\)

  

45

1

7T6

6

 

\([-1, 3]\)

\(9.13_{\ell }\)

12.54

\(3^{6} 73^{2}\)

26

0.50

6.71

0

15T47

14

 

\([-1, 2]\)

\(19.39_{g}\)

36.05

\(3^{24} 53^{6}\)

4

0.86

26.12

0

21T33

14

 

\([-1, 2]\)

\(19.39_{g}\)

31.07

\(3^{18} 17^{10}\)

2

0.86

26.12

0

42T294

15

 

\([-1, 3]\)

\(18.18_{v}\)

35.73

\(2^{12} 3^{20} 7^{12}\)

1

0.80

21.02

0

70

10

\(\sqrt{-7}\)

\([-4, 2]\)

\(22.49_{g}\)

41.21

\(2^{9} 3^{14} 7^{8}\)

1

0.90

30.75

0

42T299

21

 

\([-3, 1]\)

\(26.95_{g}\)

38.33

\(2^{18} 3^{30} 7^{16}\)

1

0.95

37.54

0

70

35

 

\([-1, 1]\)

28.79 \(_{g}\)

41.28

\(2^{30} 3^{50} 7^{28}\)

1

\(0.97_\circ \)

40.36

0

\(S_7\)

5040

 

15, –

 

40.49

\(2^*3^*5^*\)

  

35

0

7T7

6

 

\([-1, 4]\)

\(7.50_{\ell }\)

7.55

184607

 

0.33

3.27

0

14T46

6

 

\([-4, 3]\)

7.66 \(_{p}\)

17.02

\(2^{2} 7^{5} 19^{2}\)

194

0.50

5.92

0

30T565

14

 

\([-2, 4]\)

\(16.32_{p}\)

26.02

\(2^{20} 53^{8}\)

2

0.71

12.67

0

30T565

14

 

\([-4, 2]\)

\(20.24_{g}\)

30.98

\(2^{14} 71^{9}\)

46

0.86

21.06

0

42T413

14

 

\([-6, 2]\)

\(20.24_{g}\)

38.27

\(2^{20} 3^{12} 11^{10}\)

6

0.86

21.06

0

21T38

14

 

\([-1, 6]\)

\(13.12_{p}\)

22.02

\(2^{24} 3^{12} 29^{4}\)

170

0.57

7.63

0

42T412

15

 

\([-3, 5]\)

\(16.96_{p}\)

32.90

\(3^{12} 5^{5} 11^{13}\)

24

0.67

10.70

0

42T411

15

 

\([-5, 3]\)

\(16.56_{g}\)

29.92

\(2^{30} 3^{12} 17^{6}\)

3

0.80

17.19

0

70

20

 

\([-4, 2]\)

\(23.53_{g}\)

35.18

\(2^{34} 53^{12}\)

2

0.90

24.53

0

42T418

21

 

\([-3, 3]\)

\(20.24_{g}\)

33.42

\(2^{41} 3^{18} 17^{9}\)

3

0.86

21.06

0

84

21

 

\([-3, 1]\)

\(28.27_{g}\)

39.59

\(2^{38} 3^{18} 7^{16}\)

4

0.95

29.55

0

70

35

 

\([-1, 5]\)

25.92 \(_{p}\)

40.71

\(2^{61} 3^{30} 7^{28}\)

4

0.86

21.06

0

126

35

 

\([-5, 1]\)

30.23 \(_{g}\)

43.26

\(2^{54} 3^{42} 5^{30}\)

 

\(0.97_\circ \)

31.62

0

Table 6

Artin L-functions of small conductor from octic groups

\(\varvec{G}\)

\(\varvec{n_1}\)

\(\varvec{z}\)

\(\varvec{\mathfrak {d}}\)

\(\varvec{\delta _1}\)

\(\varvec{\Delta _1}\)

Pos’n

\(\varvec{\beta }\)

\(\varvec{B^\beta }\)

\(\varvec{\#}\)

\(C_8\)

8

TW

4, 8

 

11.93

\(17^*\)

  

125

198

8T1

1

\(\zeta _{8}\)

\([-4, 0]\)

\(8.84_{S}\)

17

17

1

\(1.14_\bullet \)

249.15

41

\(Q_8\)

8

TW

5, 10

 

18.24

\(2^*3^*\)

  

100

72

8T5

2

 

\([-2, 0]\)

\(26.29_{S}\)

48

\(2^{8} 3^{2}\)

2

\(1.33_\bullet \)

464.16

41

\(D_8\)

16

TW

6, 20

 

9.75

\(5^*19^*\)

  

125

6049

8T6

2

\(\sqrt{2}\)

\([-4, 0]\)

\(9.07_{q}\)

9.75

\(5 \cdot 19\)

1

1

125

2296

\(C_8{\rtimes }C_2\)

16

 

7, 24

 

9.32

\(3^*5^*\)

  

125

672

8T7

2

i

\([-4, 0]\)

\(9.07_{q}\)

15

\(3^{2} 5^{2}\)

1

1

125

75

\(QD_{16}\)

16

 

6, 20

 

10.46

\(2^*3^*\)

  

125

1664

8T8

2

\(\sqrt{-2}\)

\([-4, 0]\)

\(9.07_{q}\)

16.97

\(2^{5} 3^{2}\)

1

1

125

155

\(Q_8{\rtimes }C_2\)

16

 

9, 32

 

9.80

\(2^*3^*\)

  

100

3366

8T11

2

i

\([-4, 0]\)

\(9.07_{q}\)

10.95

\(2^{3} 3 \cdot 5\)

3

1

100

825

\(\text {SL}_2(3)\)

24

TW

5, 14

 

29.84

\(163^*\)

  

250

681

24T7

2

 

\([-2, 1]\)

\(65.51_{P}\)

163

\(163^{2}\)

1

\(1.20_\bullet \)

754.27

94

8T12

2

\(\sqrt{-3}\)

\([-4, 1]\)

\(8.09_{p}\)

12.77

163

1

0.75

62.87

78

 

32

 

11, 74

 

13.79

\(2^*5^*\)

  

125

11,886

8T15

4

 

\([-4, 0]\)

\(12.92_{q}\)

16.12

\(2^{4} 5^{2} 13^{2}\)

4

1

125

3464

 

32

 

9, 58

 

13.56

\(5^*11^*\)

  

125

766

8T16

4

 

\([-4, 0]\)

\(12.92_{q}\)

16.58

\(5^{4} 11^{2}\)

1

1

125

129

\(C_4\wr C_2\)

32

 

10, 90

 

13.37

\(2^*5^*\)

  

125

2748

8T17

\(2^{{{2}}}\)

i

\([-4, 2]\)

5.74 \(_{p}\)

8.25

\(2^{2} 17\)

6

\(0.50_\circ \)

11.18

3

 

32

 

9, 58

 

14.05

\(2^*\)

  

125

2720

8T19

4

 

\([-4, 0]\)

\(12.92_{q}\)

19.03

\(2^{17}\)

1

1

125

1282

 

32

 

17, 806

 

18.42

\(2^*3^*5^*\)

  

100

3284

8T22

4

 

\([-4, 0]\)

\(12.92_{q}\)

20.49

\(2^{4} 3^{2} 5^{2} 7^{2}\)

3

1

100

1162

\(\text {GL}_2(3)\)

48

 

7, 41

 

16.52

\(2^*43^*\)

  

100

2437

24T22

2

\(\sqrt{-2}\)

\([-4, 2]\)

5.74 \(_{v}\)

16.82

283

2

\(0.50_\circ \)

10

0

8T23

4

 

\([-4, 1]\)

\(9.07_{p}\)

9.95

\(3^{4} 11^{2}\)

4

0.75

31.62

99

\(C_2^3{\rtimes }C_7\)

56

TW

3, 4

 

17.93

\(29^*\)

  

200

28

8T25

7

 

\([-1, 0]\)

\(16.10_{q}\)

17.93

\(29^{6}\)

1

1

200

27

 

64

 

16, –

 

20.37

\(2^*5^*\)

  

125

10,317

8T26

\(4^{{{2}}}\)

 

\([-4, 2]\)

\(9.07_{p}\)

12.85

\(3^{2} 5^{2} 11^{2}\)

7

\(0.50_\circ \)

11.18

0

\(C_2\wr C_4\)

64

 

11, 206

 

19.44

\(2^*\)

  

125

2482

8T27

\(4^{{{2}}}\)

 

\([-4, 2]\)

\(9.07_{p}\)

10.60

\(5^{3} 101\)

19

0.50

11.18

1

\(C_2 \wr C_2^2\)

64

 

16, –

 

19.41

\(2^*7^*\)

  

125

11,685

8T29

\(4^{{{2}}}\)

 

\([-4, 2]\)

\(9.07_{p}\)

10.13

\(2^{4} 3^{2} 73\)

28

0.50

11.18

1

 

64

 

11, 206

 

19.44

\(2^*\)

  

125

1217

8T30

\(4^{{{2}}}\)

 

\([-4, 2]\)

\(9.07_{p}\)

14.57

\(5^{3} 19^{2}\)

3

\(0.50_\circ \)

11.18

0

 

96

 

9, 49

 

34.97

\(2^*5^*13^*\)

  

250

5520

8T32

4

 

\([-4, 1]\)

\(9.12_{g}\)

22.80

\(2^{6} 5^{2} 13^{2}\)

2

0.75

62.87

180

24T97

4

\(\sqrt{-3}\)

\([-8, 1]\)

13.19 \(_{g}\)

43.30

\(2^{6} 5^{2} 13^{3}\)

2

\(0.88_\circ \)

125.37

112

 

96

 

8, 44

 

30.01

\(2^*5^*7^*\)

  

150

791

8T33

\(6^{{{2}}}\)

 

\([-2, 2]\)

\(11.29_{p}\)

25.14

\(5^{3} 7^{4} 29^{2}\)

12

0.67

28.23

3

 

96

 

10, 92

 

27.28

\(2^*3^*31^*\)

  

110

1915

8T34

6

 

\([-2, 2]\)

\(11.29_{p}\)

22.61

\(31^{3} 67^{2}\)

64

0.67

22.96

1

\(C_2\wr D_4\)

128

 

20, –

 

22.91

\(2^*3^*13^*\)

  

125

14,369

8T35

\(4^{{{4}}}\)

 

\([-4, 2]\)

\(9.07_{p}\)

9.45

\(5^{2} 11 \cdot \ 29\)

110

0.50

11.18

9

\(C_2^3{\rtimes }F_{21}\)

168

 

5, 14

 

31.64

\(2^*73^*\)

  

200

342

8T36

7

 

\([-1, 1]\)

\(16.06_{p}\)

21.03

\(2^{6} 73^{4}\)

1

0.86

93.82

120

24T283

7

\(\sqrt{-3}\)

\([-2, 1]\)

20.23 \(_{p}\)

38.55

\(2^{6} 7^{11}\)

2

\(0.93_\circ \)

136.98

81

\(C_2\wr A_4\)

192

 

12, 700

 

37.27

\(2^*5^*7^*\)

  

250

13,649

8T38

\(4^{{{2}}}\)

 

\([-4, 2]\)

\(8.56_{v}\)

15.20

\(2^{6} 7^{2} 17\)

11

0.50

15.81

1

24T288

\(4^{{{2}}}\)

\(\sqrt{-3}\)

\([-8, 4]\)

10.84 \(_{p}\)

23.95

\(2^{6} 3 \cdot 5 \cdot 7^{3}\)

66

0.50

15.81

0

 

192

 

13, 559

 

32.35

\(2^*23^*\)

  

100

1193

8T39

\(4^{{{2}}}\)

 

\([-4, 2]\)

\(5.74_{p}\)

8.71

\(2^{4} 359\)

49

0.50

10

8

24T333

8

 

\([-8, 1]\)

15.28 \(_{g}\)

39.94

\(2^{10} 43^{6}\)

2

\(0.88_\circ \)

56.23

16

 

192

 

13, 559

 

29.71

\(2^*23^*\)

  

100

2001

8T40

\(4^{{{2}}}\)

 

\([-4, 2]\)

5.74 \(_{p}\)

13.04

\(2^{2} 5^{2} 17^{2}\)

9

\(0.50_\circ \)

10

0

24T332

8

 

\([-8, 1]\)

15.28 \(_{g}\)

29.71

\(2^{12} 23^{6}\)

1

\(0.88_\circ \)

56.23

47

 

192

 

14, 1210

 

28.11

\(2^*11^*\)

  

100

4723

12T108

\(6^{{{2}}}\)

 

\([-2, 2]\)

11.29 \(_{p}\)

20.78

\(2^{12} 3^{9}\)

5

0.67

21.54

2

8T41

\(6^{{{2}}}\)

 

\([-2, 2]\)

\(11.29_{p}\)

13.01

\(5^{3} 197^{2}\)

13

0.67

21.54

40

\(A_4\wr C_2\)

288

 

10, 178

 

32.18

\(2^*37^*\)

  

135

1362

8T42

6

 

\([-2, 3]\)

\(11.29_{p}\)

11.58

\(5^{3} 139^{2}\)

76

0.50

11.62

1

18T112

9

 

\([-3, 1]\)

17.10 \(_{g}\)

35.11

\(2^{24} 13^{6}\)

2

0.89

78.28

66

12T128

9

 

\([-3, 3]\)

\(13.59_{p}\)

22.52

\(7^{6} 233^{3}\)

56

0.67

26.32

6

24T703

6

\(\sqrt{-3}\)

\([-4, 4]\)

13.79 \(_{p}\)

32.18

\(2^{4} 37^{5}\)

1

0.67

26.32

0

\(C_2 \wr S_4\)

384

 

20, –

 

31.38

\(5^*197^*\)

  

100

6400

8T44

\(4^{{{4}}}\)

 

\([-4, 2]\)

\(5.74_{p}\)

7.53

\(5 \cdot 643\)

391

0.50

10

26

24T708

\(8^{{{2}}}\)

 

\([-8, 4]\)

13.19 \(_{p}\)

25.55

\(2^{12} 5^{2} 11^{6}\)

4

0.50

10

0

 

576

 

16, –

 

29.35

\(2^*3^*\)

  

100

2664

12T161

6

 

\([-2, 3]\)

7.60 \(_{p}\)

19.04

\(2^{8} 3^{3} 83^{2}\)

1179

0.50

10

0

8T45

6

 

\([-2, 3]\)

\(7.60_{p}\)

15.48

\(2^{14} 29^{2}\)

110

0.50

10

0

18T179

9

 

\([-3, 1]\)

\(18.84_{g}\)

29.69

\(2^{12} 5^{6} 23^{4}\)

16

0.89

59.95

121

12T165

9

 

\([-3, 3]\)

\(10.60_{p}\)

17.20

\(2^{6} 19^{3} 67^{3}\)

161

0.67

21.54

12

18T185

\(9^{{{2}}}\)

 

\([-3, 3]\)

13.74 \(_{p}\)

22.69

\(2^{12} 3^{11} 13^{3}\)

6

0.67

21.54

0

24T1504

12

 

\([-4, 4]\)

16.40 \(_{p}\)

35.03

\(2^{8} 5^{6} 31^{8}\)

51

0.67

21.54

0

 

576

 

11, 522

 

49.75

\(3^*5^*7^*\)

  

100

153

8T46

6

 

\([-2, 3]\)

\(9.66_{p}\)

19.51

\(3^{6} 5^{4} 11^{2}\)

42

0.50

10

0

12T160

6

 

\([-2, 3]\)

7.60 \(_{p}\)

27.27

\(2^{23} 7^{2}\)

11

0.50

10

0

16T1030

9

 

\([-3, 1]\)

\(18.84_{g}\)

35.40

\(2^{22} 3^{6} 13^{4}\)

6

0.89

59.95

50

18T184

9

 

\([-3, 1]\)

\(18.84_{g}\)

35.50

\(2^{12} 5^{7} 23^{4}\)

7

0.89

59.95

32

24T1506

12

 

\([-4, 4]\)

16.40 \(_{p}\)

55.16

\(2^{20} 3^{18} 5^{9}\)

4

0.67

21.54

0

36T766

9

i

\([-6, 2]\)

18.84 \(_{g}\)

58.55

\(2^{36} 7^{6}\)

3

0.89

59.95

2

\(S_4\wr C_2\)

1152

 

20, –

 

35.05

\(2^*5^*41^*\)

  

150

23,694

12T200

6

 

\([-2, 4]\)

7.60 \(_{p}\)

15.62

\(5^{3} 11^{2} 31^{2}\)

20,668

\(0.33_\circ \)

5.31

0

8T47

6

 

\([-2, 4]\)

\(7.60_{p}\)

10.51

\(2^{9} 2633\)

20,566

0.33

5.31

0

12T201

6

 

\([-4, 3]\)

7.60 \(_{p}\)

19.19

\(3^{7} 151^{2}\)

21

0.50

12.25

0

12T202

6

 

\([-4, 3]\)

7.60 \(_{p}\)

16.51

\(3^{10} 7^{3}\)

12

0.50

12.25

0

18T272

9

 

\([-3, 3]\)

9.70 \(_{p}\)

19.73

\(3^{6} 853^{3}\)

450

0.67

28.23

44

18T274

9

 

\([-3, 3]\)

\(15.95_{p}\)

27.07

\(2^{12} 5^{7} 29^{3}\)

105

0.67

28.23

3

18T273

9

 

\([-3, 3]\)

12.88 \(_{p}\)

30.86

\(2^{16} 3^{18}\)

5

\(0.67_\circ \)

28.23

0

16T1294

9

 

\([-3, 3]\)

\(10.60_{p}\)

13.16

\(43^{3} 53^{3}\)

39

0.67

28.23

295

36T1946

12

 

\([-4, 4]\)

14.77 \(_{p}\)

32.80

\(2^{10} 13^{7} 17^{6}\)

16

0.67

28.23

0

24T2821

12

 

\([-4, 4]\)

16.03 \(_{p}\)

26.48

\(2^{16} 5^{6} 41^{5}\)

1

0.67

28.23

1

36T1758

18

 

\([-6, 2]\)

20.29 \(_{g}\)

36.08

\(2^{24} 5^{9} 41^{9}\)

1

0.89

85.96

1222

Table 7

Artin L-functions of small conductor from nonic groups

\(\varvec{G}\)

\(\varvec{n_1}\)

\(\varvec{z}\)

\(\varvec{\mathfrak {d}}\)

\(\varvec{\delta _1}\)

\(\varvec{\Delta _1}\)

Pos’n

\(\varvec{\beta }\)

\(\varvec{B^\beta }\)

\(\varvec{\#}\)

\(C_9\)

9

TW

3, 4

 

13.70

\(19^*\)

  

200

48

9T1

1

\(\zeta _{9}\)

\([-3, 0]\)

\(17.02_{Q}\)

19.00

19

1

\(1.13_\bullet \)

387.85

26

\(D_9\)

18

TW

4, 8

 

12.19

\(2^*59^*\)

  

200

699

9T3

2

\(\zeta _9^+\)

\([-3, 0]\)

\(9.70_{q}\)

14.11

199

3

1.00

200

638

\(3^{1+2}_{-}\)

27

 

6, 12

 

31.18

\(7^*13^*\)

  

200

32

9T6

3

\(\sqrt{-3}\)

\([-3, 0]\)

\(30.32_{q}\)

38.70

\(7^{3} 13^{2}\)

1

1.00

200

14

\(3^{1+2}_{+}\)

27

TW

6, 12

 

50.20

\(3^*19^*\)

  

200

16

9T7

3

\(\sqrt{-3}\)

\([-3, 0]\)

\(30.32_{q}\)

64.08

\(3^{6} 19^{2}\)

1

1.00

200

12

\(3^{1+2}.2\)

54

 

7, 34

 

17.01

\(2^*3^*\)

  

200

981

9T10

6

 

\([-3, 0]\)

\(15.90_{q}\)

17.49

\(31^{5}\)

2

1.00

200

741

 

54

 

7, 34

 

16.83

\(3^*7^*\)

  

200

880

9T11

6

 

\([-3, 0]\)

\(15.90_{q}\)

19.01

\(3^{9} 7^{4}\)

1

1.00

200

805

 

54

 

8, 42

 

16.72

\(2^*3^*5^*\)

  

200

2637

9T12

3

\(\sqrt{-3}\)

\([-3, 2]\)

\(7.75_{p}\)

10.71

\(2^{2} 307\)

7

0.67

34.20

256

18T24

3

\(\sqrt{-3}\)

\([-3, 1]\)

10.03 \(_{g}\)

20.08

\(2^{2} 3^{4} 5^{2}\)

1

\(0.83_\circ \)

82.70

77

\(M_9\)

72

 

6, 20

 

29.72

\(2^*3^*\)

  

100

27

9T14

8

 

\([-1, 0]\)

\(17.50_{q}\)

31.59

\(2^{24} 3^{10}\)

1

1.00

100

26

\(C_3^2{\rtimes }C_8\)

72

TW

5, 16

 

25.41

\(2^*3^*\)

  

100

19

9T15

8

 

\([-1, 0]\)

\(17.50_{q}\)

25.41

\(2^{31} 3^{4}\)

1

1.00

100

16

\(C_3\wr C_3\)

81

 

9, 59

 

75.41

\(3^*19^*\)

  

500

131

9T17

\(3^{{{3}}}\)

\(\sqrt{-3}\)

\([-3, 3]\)

\(12.92_{q}\)

30.14

\(7^{2} 13 \cdot 43\)

22

0.50

22.36

0

\(C_3^2{\rtimes }D_6\)

108

 

11, 262

 

22.06

\(3^*23^*\)

  

150

12,002

18T55

6

 

\([-3, 1]\)

11.98 \(_{g}\)

24.38

\(2^{8} 3^{8} 5^{3}\)

4

\(0.83_\circ \)

65.07

229

9T18

6

 

\([-3, 2]\)

\(9.70_{p}\)

13.46

\(3^{3} 7^{2} 67^{2}\)

53

0.67

28.23

216

\(C_3^2{\rtimes }QD_{16}\)

144

 

8, 62

 

23.41

\(3^*7^*\)

  

100

488

9T19

8

 

\([-1, 2]\)

12.13 \(_{v}\)

25.65

\(3^{13} 7^{6}\)

1

0.75

31.62

14

18T68

8

 

\([-2, 1]\)

14.44 \(_{g}\)

25.65

\(3^{13} 7^{6}\)

1

\(0.88_\circ \)

56.23

48

\(C_3\wr S_3\)

162

 

13, 2004

 

29.89

\(3^*\)

  

200

1617

9T20

\(3^{{{3}}}\)

\(\sqrt{-3}\)

\([-3, 3]\)

\(7.75_{p}\)

11.17

\(7 \cdot 199\)

23

0.50

14.14

20

18T86

\(3^{{{3}}}\)

\(\sqrt{-3}\)

\([-3, 3]\)

8.23 \(_{v}\)

19.48

\(2^{2} 43^{2}\)

5

0.50

14.14

0

 

162

 

11, 223

 

24.90

\(2^*3^*5^*\)

  

100

597

9T21

\(6^{{{3}}}\)

 

\([-3, 3]\)

\(9.70_{p}\)

15.58

\(5^{2} 83^{3}\)

2

0.50

14.14

0

 

162

 

10, 205

 

26.46

\(3^*\)

  

100

180

9T22

\(6^{{{3}}}\)

 

\([-3, 3]\)

\(9.70_{p}\)

17.21

\(2^{6} 7^{4} 13^{2}\)

6

0.50

10

0

\(C_3^2{\rtimes }\text {SL}_2(3)\)

216

 

7, 44

 

49.57

\(349^*\)

  

100

37

9T23

8

 

\([-1, 2]\)

\(11.29_{p}\)

23.39

\(547^{4}\)

10

0.75

31.62

3

24T569

8

\(\sqrt{-3}\)

\([-2, 1]\)

\(18.99_{g}\)

38.84

\(349^{5}\)

1

0.94

74.99

20

 

324

 

17, –

 

30.64

\(2^*3^*11^*\)

  

100

1816

18T129

\(6^{{{3}}}\)

 

\([-3, 3]\)

9.34 \(_{p}\)

22.88

\(2^{9} 23^{4}\)

16

0.50

14.14

0

9T24

\(6^{{{3}}}\)

 

\([-3, 3]\)

\(9.70_{p}\)

15.84

\(3^{7} 5^{2} 17^{2}\)

399

0.50

14.14

0

 

324

 

9, 116

 

29.96

\(2^*3^*\)

  

100

107

9T25

6

 

\([-3, 3]\)

\(12.73_{p}\)

22.25

\(2^{6} 3^{8} 17^{2}\)

59

0.50

10

0

18T141

6

 

\([-3, 3]\)

9.34 \(_{p}\)

30.81

\(3^{8} 19^{4}\)

9

0.50

10

0

12T133

4

\(\sqrt{-3}\)

\([-4, 2]\)

\(11.15_{g}\)

19.34

\(2^{6} 3^{7}\)

1

0.75

31.62

10

12T132

\(4^{{{2}}}\)

\(\sqrt{-3}\)

\([-4, 2]\)

11.15 \(_{g}\)

33.50

\(2^{6} 3^{9}\)

1

0.75

31.62

0

18T142

12

 

\([-3, 3]\)

\(19.68_{p}\)

30.57

\(2^{18} 3^{26}\)

3

0.75

31.62

2

\(C_3^2{\rtimes }\text {GL}_2(3)\)

432

 

10, 206

 

27.88

\(3^*11^*\)

  

76

453

9T26

8

 

\([-1, 2]\)

\(11.54_{g}\)

17.59

\(2^{6} 523^{3}\)

14

0.75

25.74

17

18T157

8

 

\([-2, 2]\)

12.14 \(_{p}\)

19.04

\(3^{7} 53^{4}\)

16

0.75

25.74

3

24T1334

16

 

\([-2, 1]\)

\(21.27_{g}\)

26.68

\(3^{26} 11^{10}\)

1

0.94

57.98

134

\(S_3\wr C_3\)

648

 

14, 3706

 

33.56

\(2^*5^*13^*\)

  

150

1677

18T197

\(6^{{{2}}}\)

 

\([-3, 3]\)

9.70 \(_{v}\)

19.71

\(7^{4} 29^{3}\)

1

0.50

12.25

0

18T202

6

 

\([-4, 3]\)

9.70 \(_{v}\)

27.73

\(2^{6} 3^{9} 19^{2}\)

49

0.50

12.25

0

9T28

6

 

\([-3, 4]\)

\(9.70_{p}\)

12.20

\(3^{8} 503\)

335

0.33

5.31

0

12T176

8

 

\([-4, 2]\)

\(12.05_{g}\)

21.75

\(11^{4} 43^{4}\)

4

0.75

42.86

57

36T1102

12

 

\([-4, 3]\)

15.58 \(_{v}\)

29.90

\(2^{6} 5^{10} 13^{8}\)

1

0.75

42.86

2

18T206

12

 

\([-3, 4]\)

\(15.23_{v}\)

21.99

\(2^{6} 7^{10} 29^{4}\)

10

0.67

28.23

8

24T1539

8

\(\sqrt{-3}\)

\([-8, 4]\)

17.07 \(_{v}\)

45.71

\(2^{14} 3^{19}\)

3

0.75

42.86

0

 

648

 

13, 2206

 

40.81

\(2^*3^*17^*\)

  

200

838

9T29

6

 

\([-3, 3]\)

\(12.73_{p}\)

16.62

\(2^{8} 7^{2} 41^{2}\)

31

0.50

14.14

0

18T223

6

 

\([-3, 3]\)

9.70 \(_{v}\)

30.14

\(2^{4} 5^{2} 37^{4}\)

71

0.50

14.14

0

24T1527

4

\(\sqrt{-3}\)

\([-4, 2]\)

\(12.05_{g}\)

32.34

\(2^{2} 3^{7} 5^{3}\)

18

0.75

53.18

43

12T175

4

\(\sqrt{-3}\)

\([-4, 4]\)

\(7.60_{p}\)

9.23

\(11 \cdot 659\)

164

0.50

14.14

14

36T1131

6

\(\sqrt{-3}\)

\([-6, 6]\)

11.95 \(_{v}\)

36.04

\(2^{2} 3^{8} 17^{4}\)

6

0.50

14.14

0

36T1237

12

 

\([-3, 3]\)

17.78 \(_{p}\)

44.72

\(2^{22} 3^{7} 17^{8}\)

1

0.75

53.18

6

18T219

12

 

\([-3, 3]\)

14.05 \(_{v}\)

33.23

\(3^{17} 107^{5}\)

5

0.75

53.18

65

24T1540

8

\(\sqrt{-3}\)

\([-8, 4]\)

17.07 \(_{v}\)

49.37

\(2^{10} 3^{15} 7^{4}\)

2

0.75

53.18

1

 

648

 

13, 1322

 

30.37

\(2^*269^*\)

  

200

4001

9T30

6

 

\([-3, 3]\)

\(9.70_{p}\)

10.67

\(11^{2} 23^{3}\)

3

0.50

14.14

1

18T222

6

 

\([-3, 3]\)

9.70 \(_{v}\)

23.27

\(31^{3} 73^{2}\)

57

0.50

14.14

0

12T178

8

 

\([-4, 2]\)

\(12.05_{g}\)

18.27

\(2^{10} 59^{4}\)

10

0.75

53.18

327

12T177

\(8^{{{2}}}\)

 

\([-4, 2]\)

12.05 \(_{g}\)

24.98

\(2^{10} 23^{6}\)

22

0.75

53.18

173

36T1121

6

\(\sqrt{3}\)

\([-6, 6]\)

11.95 \(_{v}\)

31.61

\(2^{8} 7^{2} 43^{3}\)

91

0.50

14.14

0

36T1123

12

 

\([-3, 3]\)

17.78 \(_{p}\)

28.87

\(2^{35} 5^{10}\)

2

0.75

53.18

71

18T218

12

 

\([-3, 3]\)

\(14.05_{v}\)

18.33

\(2^{10} 269^{5}\)

1

0.75

53.18

453

\(S_3 \wr S_3\)

1296

 

22, –

 

36.26

\(2^*3^*\)

  

200

12,152

18T320

6

 

\([-3, 4]\)

8.80 \(_{p}\)

14.80

\(5 \cdot 23^{3} 173\)

8562

0.33

5.85

0

18T312

6

 

\([-4, 3]\)

8.79 \(_{p}\)

17.45

\(3^{5} 11^{2} 31^{2}\)

343

0.50

14.14

0

9T31

6

 

\([-3, 4]\)

\(9.70_{p}\)

10.38

\(31^{2} 1303\)

10,036

0.33

5.85

0

18T303

6

 

\([-4, 3]\)

8.79 \(_{p}\)

18.34

\(5^{5} 23^{3}\)

4

0.50

14.14

0

12T213

8

 

\([-4, 4]\)

\(11.29_{p}\)

13.38

\(5^{2} 7^{4} 131^{2}\)

397

0.50

14.14

3

24T2895

8

 

\([-4, 2]\)

12.79 \(_{g}\)

30.65

\(2^{8} 3^{4} 5^{6} 7^{4}\)

77

0.75

53.18

230

18T315

12

 

\([-3, 4]\)

\(13.59_{p}\)

22.81

\(2^{10} 23^{4} 37^{5}\)

72

0.67

34.20

105

36T2216

12

 

\([-3, 4]\)

13.79 \(_{p}\)

29.13

\(2^{10} 3^{17} 41^{4}\)

78

0.67

34.20

11

36T2305

12

 

\([-4, 3]\)

15.14 \(_{p}\)

31.73

\(2^{18} 331^{5}\)

58

0.75

53.18

151

36T2211

12

 

\([-6, 6]\)

14.64 \(_{p}\)

37.29

\(2^{16} 5^{8} 7^{10}\)

55

0.50

14.14

0

36T2214

12

 

\([-4, 3]\)

15.14 \(_{p}\)

32.07

\(2^{22} 3^{24}\)

11

0.75

53.18

136

24T2912

16

 

\([-8, 4]\)

18.82 \(_{p}\)

35.12

\(5^{12} 23^{12}\)

4

0.75

53.18

40

Our second result unconditionally identifies initial segments:

Theorem 8.2

For 144 Galois types \((G,\chi )\), Tables 3, 4, 5, 6 and 7 identify a non-empty initial segment \(\mathcal {L}(G,\chi ;B^\beta )\), and in particular identify the minimal root conductor \(\delta _1(G,\chi )\).

8.2 Tables detailing results on lower bounds and initial segments

Our tables are organized by the standard doubly-indexed lists of transitive permutation groups mTj, with degrees m running from 2 through 9. Within a degree, the blocks of rows are indexed by increasing j. There is no block to print if mTj has no faithful irreducible characters. For example, there is no block to print for groups having noncyclic center, such as \(4T2 = V = C_2 \times C_2\) or \(8T9 = D_4 \times C_2\). Also the block belonging to mTj is omitted if the abstract group G underlying mTj has appeared earlier. For example \(G=S_4\) has four transitive realization in degrees \(m \le 8\), namely 4T5, 6T7, 6T8, and 8T14; there is correspondingly a 4T5 line on our tables, but no 6T7, 6T8, or 8T14 lines.

8.2.1 Top row of the G-block

The top row in the G-block is different from the other rows, as it gives information corresponding to the abstract group G. Instead of referring to a faithful irreducible character, as the other lines do, many of its entries are the corresponding quantities for the regular character \(\phi _G\). The first four entries are a common name for the group G (if there is one), the order \(\phi _G(e) = |G|\), the symbol TW if G is known to have the universal tame-wild property as defined in [11], and finally kN. Here, k is the size of the rational character table, and N is number of vertices of the polytope \(P_G\) discussed in Sect. 6.2, or a dash if we did not compute N. The last four entries are the smallest root discriminant of a Galois G field, the factored form of the corresponding discriminant, a cutoff B for which the set \(\mathcal {K}(G;B)\) is known, and the size \(|\mathcal {K}(G;B)|\).

8.2.2 Remaining rows of the G-block

Each remaining line of the G-block corresponds to a type \((G,\chi )\). However the number of rows in the G-block is typically substantially less than the number of faithful irreducible characters of G, as we list only one representative of each \({{\mathrm{Gal}}}(\overline{\mathbf {Q}}/\mathbf {Q}) \times \text{ Out }(G)\) orbit of such characters. As an example, \(S_6\) has eleven characters, all rational. Of the nine which are faithful, there are three which are fixed by the nontrivial element of \(\text{ Out }(S_6)\) and the others form three two-element orbits. Thus the \(S_6\)-block has six rows. In general, the information on a \((G,\chi )\) row comes in three parts, which we now describe in turn.

First four columns The first column gives the lexicographically first permutation group mTj for which the corresponding permutation character has \(\chi \) as a rational constituent. Then \(n_1=\chi _1(e)\) is the degree of an absolutely irreducible character \(\chi _1\) such that \(\chi \) is the sum of its conjugates. The number \(n_1\) is superscripted by the size of the \({{\mathrm{Out}}}(G)\) orbit of \(\chi \), in the unusual case when this orbit size is not 1. Next, the complex number z is a generator for the field generated by the values of the character \(\chi _1\), with no number printed in the common case that \(\chi _1\) is rational-valued. The last entry gives the interval , where and \(\widehat{\chi }\) are the numbers introduced in the beginning of Sect. 5. In the range presented, the data of mTj, \(n_1\), z, and suffice to distinguish Galois types \((G,\chi )\) from each other.

Middle four columns The next four columns focus on minimal root conductors. In the first entry, \(\mathfrak {d}\) is the best conditional lower bound we obtained for root conductors, and the subscript \(i \in \{\ell ,s,q,g,p,v\}\) gives information on the corresponding auxiliary character \(\phi \). The first four possibilities refer to the methods of Sect. 5, namely linear, square, quadratic, and Galois. The last two, p and v, indicate a permutation character and a character coming from a vertex of the polytope \(P_G\). The best \(\phi \) of the ones we inspect is always at a vertex, except in the three cases on Table 4 where \(*\) is appended to the subscript. Capital letters S, Q, G, P, and V also appear as subscripts. These occur only for groups marked with TW, and indicate that the tame-wild principle improved the lower bound. For most groups with fifteen or more classes, it was prohibitive to calculate all vertices, and the best of the other methods is indicated.

When the second entry is in roman type, it is the minimal root conductor and the third entry is the minimal conductor in factored form. When the second entry is in italic type, then it is the smallest currently known root conductor. The fourth entry gives the position of the source number field on the complete list ordered by Galois root discriminant. This information lets readers obtain further information from [10], such as a defining polynomial and details on ramification.

Last three columns The quantity \(\beta \) is the exponent we are using to pass from Galois number fields to Artin representations. Writing \(\widehat{\underline{\alpha }}= \widehat{\underline{\alpha }}(G,\chi ,\phi _G)\) and \(\underline{\alpha }= \underline{\alpha }(G,\chi ,\phi _G)\), one has the universal relation \(\widehat{\underline{\alpha }}\le \underline{\alpha }\). When equality holds then the common number is printed. To indicate that inequality holds, an extra symbol is printed. When we know that G satisfies TW then we can use the larger exponent and \(\underline{\alpha }_\bullet \) is printed. Otherwise we use the smaller exponent and \(\widehat{\underline{\alpha }}_\circ \) is printed. The column \(B^\beta \) gives the corresponding upper bound on our complete list of root conductors. Finally the column \(\#\) gives \(|\mathcal {L}(G,\chi ; B^\beta )|\), the length of the complete list of Artin L-functions we have identified. For the L-functions themselves, we refer to [14].

9 Discussion of tables

In this section, we discuss four topics, each of which makes specific reference to parts of the tables of the previous section. Each of the topics also serves the general purpose of making the tables more readily understandable.

9.1 Comparison of first Galois root discriminants and root conductors

Suppose first, for notational simplicity, that G is a group for which all irreducible complex characters take rational values only. When one fixes \(K^\mathrm{gal}\) with \({{\mathrm{Gal}}}(K^\mathrm{gal}/\mathbf {Q}) \cong G\) and lets \(\chi \) runs over all the irreducible characters of G, the root discriminant \(\delta _\mathrm{Gal}\) is just the weighted multiplicative average \(\prod _\chi = {\delta _\chi ^{\chi (e)^2/|G|}}\). Deviation of a root conductor \(\delta _\chi \) from \(\delta _\mathrm{Gal}\) is caused by nonzero values of \(\chi \). When \(\chi (e)\) is large and is small, \(\delta _\chi \) is necessarily close to \(\delta _\mathrm{Gal}\). One can therefore think generally of \(\delta _\mathrm{Gal}\) as a first approximation to \(\delta _\mathrm{\chi }\). The general principle of \(\delta _\mathrm{Gal}\) approximating \(\delta _\mathrm{\chi }\) applies to groups G with irrational characters as well.

Our first example of \(S_5\) illustrates both how the principle \(\delta _\mathrm{Gal} \approx \delta _{\chi }\) is reflected in the tables, and how it tends to be somewhat off in the direction that \(\delta _\mathrm{Gal} > \delta _{\chi }\). For a given \(K^\mathrm{gal}\), the variance of \(\delta _\chi \) about its \(\delta _\mathrm{Gal}\) is substantial and depends on the details of the ramification in \(K^\mathrm{gal}\). There are many \(K^\mathrm{gal}\) with root discriminant near the minimal root discriminant, all of which are possible sources of minimal root conductors. It is therefore expected that the minimal conductors \(\delta _1(S_5,\chi ) = \min \delta _\chi \) printed in the table, 6.33, 18.72, 16.27, 17.78, and 18.18, are substantially less than the printed minimal root discriminant \(\delta _1(S_5,\phi _{120}) \approx 24.18\). As groups G get larger, one can generally expect tighter clustering of the \(\delta _1(G,\chi )\) about \(\delta _1(G,\phi _G)\). One can see the beginning of this trend in our partial results for \(S_6\) and \(S_7\).

9.2 Known and unknown minimal root conductors

Our method of starting with a complete list of Galois fields is motivated by the principle from the previous subsection that the Galois root discriminant \(\delta _\mathrm{Gal}\) is a natural first approximation to \(\delta _\chi \). Indeed, as the tables show via nonzero entries in the \(\#\) column, this general method suffices to obtain a non-empty initial segment for most \((G,\chi )\). As our focus is primarily on the first root conductor \(\delta _1 = \delta _1(G,\chi )\), we do not pursue larger initial segments in these cases.

When the initial segment from our general method is empty, as reported by a 0 in the \(\#\) column, we aim to nonetheless present the minimal root conductor \(\delta _1\). Suppose there are subgroups \(H_m \subset H_k \subseteq G\), of the indicated indices, such that a multiple of the character \(\chi \) of interest is a difference of the corresponding permutation characters: \(c \chi = \phi _m - \phi _k\). Suppose one has the complete list of all degree m fields corresponding to the permutation representation of G on \(G/H_m\) and root discriminant \(\le B\). Then one can extract the complete list of \(\mathcal {L}(G,\chi ;B^{m/(m-k)})\) of desired Artin L-functions.

For example, consider \(\chi _5\), the absolutely irreducible 5-dimensional character of \(A_6\). The permutation character for a corresponding sextic field decomposes \(\phi _6 = 1+\chi _5\), and so the discriminant of the sextic field equals the conductor of \(\chi _5\). As an example with \(k>1\), consider the 6-dimension character \(\chi _6\) for \(C_3\wr C_3 = 9T17\), which is the sum of a three-dimensional character and its conjugate. The nonic field has a cubic subfield, and the characters are related by \(\phi _9 = \phi _3 + \chi _6\). In terms of conductors, \(D_9 = D_3 \cdot D_{\chi _6}\), where \(D_9\) and \(D_3\) are field discriminants. So, we can determine the minimal conductor of an L-function with type \((C_3\wr C_3,\chi _6)\) from a sufficiently long complete list of nonic number fields with Galois group \(C_3\wr C_3\).

This method, applied to both old and newer lists presented in [10], accounts for all but one of the \(\delta _1\) reported in Roman type on the same line as a 0 in the \(\#\) column. The remaining case of an established \(\delta _1\) is for the type \((\mathrm {GL}_3(2),\chi _7)\). The group \(\mathrm {GL}_3(2)\) appears on our tables as 7T5. The permutation representation 8T37 has character \(\chi _7+1\). Here the general method says that \(\mathcal {L}(\mathrm {GL}_3(2),\chi _7; 26.12)\) is empty. It is prohibitive to compute the first octic discriminant by searching among octic polynomials. In [9] we carried out a long search of septic polynomials, examining all local possibilities giving an octic discriminant at most 30. This computation shows that \(|\mathcal {L}(\mathrm {GL}_3(2),\chi _7; 48.76)| = 25\) and in particular identifies \(\delta _1 = 21^{8/7} \approx 32.44\).

The complete lists of Galois fields for a group first appearing in degree m were likewise computed by searching polynomials in degree m, targeting for small \(\delta _\mathrm{Gal}\). This single search can give many first root conductors at once. For example, the largest groups on our octic and nonic tables are \(S_4 \wr S_2 = 8T47\) and \(S_3 \wr S_3 = 9T31\). In these cases, minimal root conductors were obtained for 5 of the 10 and 7 of the 12 faithful \(\chi \) respectively. Searches adapted to a particular character \(\chi \) as in the previous paragraph can be viewed as a refinement of our method, with targeting being not for small \(\delta _\mathrm{Gal}\) but instead for small \(\delta _\chi \). Many of the italicized entries in the column \(\delta _1\) seem improvable to known minimal root conductors via this refinement.

9.3 The ratio \(\delta _1/{\mathfrak {d}}\)

In all cases in the table, \(\delta _1>{\mathfrak {d}}\). Thus, as expected, we did not encounter a contradiction to the Artin conjecture or the Riemann hypothesis. In some cases in the table, the ratio \(\delta _1/{\mathfrak {d}}\) is quite close to 1. As two series of examples, consider \(S_m\) with its reflection character \(\chi _{m-1} = \phi _m-1\), and \(D_m\) and the sum \(\chi \) of all its faithful 2-dimensional characters. Then these ratios are as follows:
$$\begin{aligned} \begin{array}{r|llllllll} m &{} \;\; 2 &{} \;\; 3 &{} \; \;4 &{} \; \; 5 &{} \; \; 6 &{} \; \; 7 &{} \; \;8 &{} \; \;9 \\ \hline \delta _1/{\mathfrak {d}} \text{ for } (S_m,\chi _{m-1}) &{} 1.00 &{} 1.02 &{} 1.2 &{} 1.005 &{} 1.1 &{} 1.007 \\ \delta _1/{\mathfrak {d}} \text{ for } (D_m,\chi ) &{} &{} &{} 1.09 &{} 1.02 &{} 1.23 &{} 1.006 &{} 1.07 &{} 1.45 \\ \end{array} \end{aligned}$$
In the cases with the smallest ratios, the transition from no L-functions to many L-functions is commonly abrupt. For example, in the case \((S_7,\chi _6)\) the lower bound is \({\mathfrak {d}} \approx 7.50\) and the first seven rounded root conductors are 7.55, 7.60, 7.61, 7.62, 7.64, 7.66, and 7.66.

When the translation from no L-functions to many L-functions is not abrupt, but there is an L-function with outlyingly small conductor, again \(\delta _1/{\mathfrak {d}}\) may be quite close to 1. As an example, for \((8T25,\chi _7)\), one has \({\mathfrak {d}} \approx 16.10\) and \(\delta _1 = 29^{6/7} \approx 17.93\) yielding \(\delta _1/{\mathfrak {d}} \approx 1.11\). However in this case the next root conductor is \(\delta _2 = 113^{6/7} \approx 57.52\), yielding \(\delta _2/{\mathfrak {d}} \approx 3.57\). Thus the close agreement is entirely dependent on the L-function with outlyingly small conductor. Even the second root conductor is somewhat of an outlier as the next three conductors are 71.70, 76.39, and 76.39, so that already \(\delta _3/{\mathfrak {d}} \approx 4.45\).

There are many \((G,\chi )\) in the table for which the ratio \(\delta _1/{\mathfrak {d}}\) is around 2 or 3. There is some room for improvement in our analytic lower bounds, for example changing the test function (3.1), varying \(\phi \) over all of \(P_G\), or replacing the exponent \(\widehat{\underline{\alpha }}\) with the best possible exponent b. However examples like the one in the previous paragraph suggest to us that in many cases the resulting increase in \(\mathfrak {d}\) towards \(\delta _1\) would be very small.

9.4 Multiply minimal fields

Tables 3, 4, 5, 6 and 7 make implicit reference to many Galois number fields, and all necessary complete lists are accessible from the database [10]. Table 8 presents a small excerpt from this database by giving six polynomials f(x). For each f(x), Table 8 first gives the Galois group G and the root discriminant \(\delta \) of the splitting field \(K^\mathrm{gal}\). We are highlighting these particular Galois number fields \(K^\mathrm{gal}\) here because they are multiply minimal: they each give rise to the minimal root conductor for at least two different rationally irreducible characters \(\chi \). The degrees of these characters are given in the last column of Table 8.
Table 8

Invariants and defining polynomials for Galois number fields giving rise to minimal root discriminants for at least two rationally irreducible characters \(\chi \)

Further information on the characters \(\chi \) is given in Tables 3, 4, 5, 6 and 7. An interesting point, evident from repeated 1’s in the G-block on these tables, is that five of the six fields \(K^\mathrm{gal}\) are also first on the list of G fields ordered by root discriminant. The exception is the \(S_6\) field on Table 8, which is only sixth on the list of Galois \(S_6\) fields ordered by root discriminant.

10 Lower bounds in large degrees

In this section, we continue our practice of assuming the Artin conjecture and Riemann hypothesis for the relevant L-functions. For n a positive integer, let \(\Delta _1(n)\) be the smallest root discriminant of a degree n field. As illustrated by Fig. 1, one has,
$$\begin{aligned} \liminf _{n \rightarrow \infty } \Delta _1(n) \ge \Omega \approx 44.7632. \end{aligned}$$
(10.1)
Now let \(\delta _1(n)\) be the smallest root conductor of an absolutely irreducible degree n Artin representation. Theorem 4.2 of [24] uses the quadratic method to conclude that \(\delta _1(n) \ge 6.59 e^{-(13278.42/n)^2}\). If one repeats the argument there without concerns for effectivity, one gets
$$\begin{aligned} \liminf _{n \rightarrow \infty } \delta _1(n) \ge \sqrt{\Omega } \approx 6.6905. \end{aligned}$$
(10.2)
The contrast between (10.1) and (10.2) is striking, and raises the question of whether \(\sqrt{\Omega }\) in (10.2) can be increased at least part way to \(\Omega \).

10.1 The constant \(\Omega \) as a limiting lower bound.

The next corollary makes use of the extreme character values and \(\widehat{\chi }\) introduced at the beginning of Sect. 5. It shows that if one restricts the type, then one can indeed increase \(\sqrt{\Omega }\) all the way to \(\Omega \). We formulate the corollary in the context of rationally irreducible characters, to stay in the main context we have set up. However via (2.4), it can be translated to a statement about absolutely irreducible characters.

Corollary 10.1

Let \((G_k,\chi _k)\) be a sequence of rationally irreducible Galois types of degree \(n_k = \chi _k(e)\) . Suppose that the number of irreducible constituents \((\chi _k,\chi _k)\) is bounded, \(n_k \rightarrow \infty \), and either
  1. A:

    , or

     
  2. B:

    \(\widehat{\chi }_k/n_k \rightarrow 0\).

     
Then, assuming the Artin conjecture and Riemann hypothesis for relevant L-functions,
$$\begin{aligned} \liminf _{k \rightarrow \infty } \delta _1(G_k,\chi _k) \ge \Omega . \end{aligned}$$
(10.3)

Proof

For Case A, Theorem 4.2 using a linear auxiliary character as in (5.1) says
For Case B, Theorem 4.2 using a Galois auxiliary character as in (5.4) says
$$\begin{aligned} \delta _1(G_k,\chi _k) \ge M \left( |G_k| , 0 ,(\chi _k,\chi _k) \right) ^{1 - {\widehat{\chi }}/{n_k}}. \end{aligned}$$
In both cases, the first argument of M tends to infinity, the second argument does not matter, the third argument does not matter either by boundedness, and the exponent tends to 1. By (3.2), these right sides thus have an infimum limit of at least \(\Omega \), giving the conclusion (10.3). \(\square \)

For the proof of Case B, the square auxiliary character would work equally well through (5.2). Also (10.1), (10.2), and Corollary 10.1 could all be strengthened by considering the placement of complex conjugation. For example, when restricting to the totally real case \(c=e\), the \(\Omega \)’s in (10.1), (10.2), and (10.3) are simply replaced by \(\Theta \approx 215.3325\).

10.2 Four contrasting examples

Many natural sequences of types are covered by either Hypothesis A or Hypothesis B, but some are not. Table 9 summarizes four sequences which we discuss together with some related sequences next.
Table 9

Four sequences of types, with Corollary 10.1 applicable to the first three

\(\varvec{G_k}\)

\(\varvec{\chi _k}\)

\(\varvec{\widehat{\chi }_k}\)

\(\varvec{n}\)

\(\varvec{|G|}\)

\(\varvec{A}\)

\(\varvec{B}\)

\(\mathrm {PGL}_2(k)\)

\(\text{ Steinberg }\)

1

1

k

\(k^3-k\)

\(\checkmark \)

\(\checkmark \)

\(S_{k}\)

\(\text{ Reflection }\)

1

\(k-3\)

\(k-1\)

k!

\(\checkmark \)

 

\(2_\epsilon ^{1+2k}\)

\(\text{ Spin }\)

\(2^k\)

0

\(2^k\)

\(2^{1+2k}\)

 

\(\checkmark \)

\(2^k.S_k\)

\(\text{ Reflection }\)

k

\(k-2\)

k

\(2^k\) k!

  

10.2.1 The group \(\mathrm {PGL}_2(k)\) and its characters of degree \(k-1\), k, and \(k+1\).

In the sequence \((\mathrm {PGL}_2(k),\chi _k)\) from the first line of Table 9, the index k is restricted to be a prime power. The permutation character \(\phi _{k+1}\) arising from the natural action of \(\mathrm {PGL}_2(k)\) on \(\mathbb {P}^1(\mathbb {F}_k)\) decomposes as \(1+\chi _k\) where \(\chi _k\) is the Steinberg character. Table 9 says that the ratios and \(\widehat{\chi }_k/n_k\) are both 1 / k, so Corollary 10.1 applies through both Hypotheses A and B.

The conductor of \(\chi _k\) is the absolute discriminant of the degree \(k+1\) number field with character \(\phi _{k+1}\). Thus, in this instance, (10.3) is already implied by the classical (10.1). However, the other nonabelian irreducible characters \(\chi \) of \(\mathrm {PGL}_2(k)\) behave very similarly to \(\chi _k\). Their dimensions are in \(\{k-1,k,k+1\}\) and their values besides \(\chi (e)\) are all in \([-2,2]\). So suppose for each k, an arbitrary nonabelian rationally irreducible character \(\chi _k\) of \(\mathrm {PGL}_2(k)\) were chosen, in such a way that the sequence \((\chi _k,\chi _k)\) is bounded. Then Corollary 10.1 would again apply through both Hypotheses A and B. But now the \(\chi _k\) are not particularly closely related to permutation characters.

10.2.2 The group \(S_{k}\) and its canonical characters

As with the last example, the permutation character \(\phi _{k}\) arising from the natural action of \(S_{k}\) on \(\{1,\ldots ,k\}\) decomposes as \(1 + \chi _k\) where \(\chi _k\) is the reflection character with degree \(k-1\). The second line of Table 9 shows that Corollary 10.1 applies through Hypothesis A. In fact, using the linear auxiliary character underlying Hypothesis A here is essential; the limiting lower bound coming from the square or quadratic auxiliary characters is \(\sqrt{\Omega }\), and this lower bound is just 1 from the Galois auxiliary character.

Again in parallel to the previous example, the familiar sequence \((S_k,\chi _k)\) of types needs to be modified to make it a good illustration of the applicability of Corollary 10.1. Characters of \(S_k\) are most commonly indexed by partitions of k, with \(\chi _{(k)} =1\), \(\chi _{(k-1,1)}\) being the reflection character, and \(\chi _{(1,1,\ldots ,1,1)}\) being the sign character. However an alternative convention is to include explicit reference to the degree k and then omit the largest part of the partition, so that the above three characters have the alternative names \(\chi _{k,()}\), \(\chi _{k,(1)}\), and \(\chi _{k,(1,\ldots ,1,1)}\). With this convention, one can prove that for any fixed partition \(\mu \) of a positive integer m, the sequence of types \((G_k,\chi _{k,\mu })\) satisfies Hypothesis A but not B.

The case of general \(\mu \) is well represented by the two cases where \(m=2\). In these two cases, information in the same format as Table 9 is
Let \(X_{k,m}\) be the \(S_k\)-set consisting of m-tuples of distinct elements of \(\{1,\ldots ,k\}\). Then its permutation character \(\phi _{k,m}\) decomposes into \(\chi _{k,\mu }\) with \(\mu \) a partition of an integer \(\le m\). These formulas are uniform in k, as in
$$\begin{aligned} \phi _{k,2} = \chi _{k,(1,1)} + \chi _{k,(2)} + 2 \chi _{k,(1)} + \chi _{k,()}. \end{aligned}$$
For \(\mu \) running over partitions of a large integer m, the characters \(\chi _{k,\mu }\) can be reasonably regarded as quite far from permutation characters, and they thus serve as a better illustration of Corollary 10.1. The sequences \((S_k,\chi _{k,\mu })\) satisfy Hypothesis A but not B, because \(n_k\) and \(\widehat{\chi }_k\) grow polynomially as \(k^m\), while grows polynomially with degree \(<m\).

10.2.3 The extra-special group \(2_\epsilon ^{1+2k}\) and its degree \(2^k\) character

Fix \(\epsilon \in \{+,-\}\). Let \(G_k\) be the extra-special 2-group of type \(\epsilon \) and order \(2^{1+2k}\), so that \(2^{1+2}_+\) and \(2^{1+2}_{-}\) are the dihedral and quaternion groups respectively. These groups each have exactly one irreducible character of degree larger than 1, this degree being \(2^k\). There are just three character values, \(-2^k\), 0, and \(2^k\). For these two sequences, Corollary 10.1 again applies, but now only through Hypothesis B.

10.2.4 The Weyl group \(2^k.S_k\) and its degree k reflection character

The Weyl group \(W(B_k) \cong 2^k.S_k\) of signed permutation matrices comes with its defining degree k character \(\chi _k\). Here, as indicated by the fourth line of Table 9, neither hypothesis of Corollary 10.1 applies.

However the conclusion (10.3) of Corollary 10.1 continues to hold as follows. Relate the character \(\chi _k\) in question to the two standard permutation characters of \(2^k.S_k\) via \(\phi _{2k} = \phi _k + \chi _k\). For a given \(2^k.S_k\) field, \(D_{\Phi _{2k}}=D_{\Phi _k}D_{\mathcal {X}_k}\). But, since \(\Phi _k\) corresponds to an index 2 subfield of the degree 2k number field for \(\Phi _{2k}\), we have \(D_{\Phi _k}^2\mid D_{\Phi _{2k}}\). Combining these we get \(D_{\Phi _k} \mid D_{\mathcal {X}_k}\) and hence \(\delta _{\Phi _k} < \delta _{\mathcal {X}_k}\). So (10.1) implies (10.3).

10.3 Concluding speculation

As we have illustrated in Sects. 10.2.110.2.3, both Hypothesis A and Hypothesis B are quite broad. This breadth, together with the fact that the conclusion (10.3) still holds for our last sequence, raises the question of whether (10.3) can be formulated more universally. While the evidence is far from definitive, we expect a positive answer. Thus we expect that the first accumulation point of the numbers \(\delta _1(G,\chi )\) is at least \(\Omega \), where \((G,\chi )\) runs over all types with \(\chi \) irreducible. Phrased differently, we expect that the first accumulation point of the root conductors of all irreducible Artin L-functions is at least \(\Omega \).

Declarations

Acknowledgements

DPR’s work on this paper was supported by Grant #209472 from the Simons Foundation and Grant DMS-1601350 from the National Science Foundation.

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Authors’ Affiliations

(1)
School of Mathematical and Statistical Sciences, Arizona State University
(2)
Division of Science and Mathematics, University of Minnesota-Morris

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