Artin L-functions to almost monomial Galois groups

TL;DR

This paper investigates the holomorphic properties of Artin L-functions for Galois extensions with almost monomial Galois groups, showing under certain conditions that all such functions are holomorphic at specific points.

Contribution

It introduces the concept of almost monomial groups and proves a holomorphicity result for Artin L-functions in this context, expanding understanding of their analytic behavior.

Findings

All Artin L-functions are holomorphic at certain points for almost monomial Galois groups.

Provides examples and basic properties of almost monomial groups.

Establishes conditions under which Artin L-functions have no common zeros at specified points.

Abstract

If $K/\mathbb Q$ is a finite Galois extension with an almost monomial Galois group and if $s_0\in\mathbb C\setminus\{1\}$ is not a common zero for any two Artin L-functions associated to distinct complex irreducible characters of the Galois group then all Artin L-functions of $K/\mathbb Q$ are holomorphic at $s_0$. We present examples and basic properties of almost monomial groups.