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## On holomorphic Artin L-functions

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Author(s) : Florin Nicolae

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 09-2013

MSC 2000

11R42 Zeta functions and $L$-functions of number fields

Abstract :
Let $K/\Q$ be a finite Galois extension, $s_0\in \C\setminus \{1\}$, $Hol(s_0)$ the semigroup of Artin L-functions holomorphic at $s_0$. We present criteria for Artin's holomorphy conjecture in terms of the semigroup $Hol(s_0)$. We conjecture that Artin's L-functions are holomorphic at $s_0$ if and only if $Hol(s_0)$ is factorial. We prove this if $s_0$ is a zero of an L-function associated to a linear character of the Galois group.

Keywords : Artin L-function; Artin's holomorphy conjecture