On holomorphic Artin L-functions

TL;DR

This paper explores conditions under which Artin L-functions are holomorphic at a point, linking the holomorphicity to the factoriality of a semigroup, with implications for Galois groups and low-dimensional L-functions.

Contribution

It establishes a new criterion connecting the holomorphicity of Artin L-functions at a point to the factoriality of their associated semigroup, especially for almost monomial Galois groups.

Findings

Artin L-functions are holomorphic at s_0 iff the semigroup of such functions is factorial for almost monomial Galois groups.

The criterion applies to zeros of irreducible L-functions of dimension ≤ 2 without Galois group restrictions.

Provides a new algebraic perspective on the holomorphicity of Artin L-functions at specific points.

Abstract

Let $K/\mathbb Q$ be a finite Galois extension, $s_0\in \mathbb C\setminus \{1\}$, ${\it Hol}(s_0)$ the semigroup of Artin L-functions holomorphic at $s_0$. If the Galois group is almost monomial then Artin's L-functions are holomorphic at $s_0$ if and only if $ {\it Hol}(s_0)$ is factorial. This holds also if $s_0$ is a zero of an irreducible L-function of dimension $\leq 2$, without any condition on the Galois group.