On holomorphic Artin L-functions

Abstract

Let $$K/\mathbb {Q}$$ be a finite Galois extension, $$s_0\in \mathbb {C}{\setminus } \{1\}$$ , $${Hol}(s_0)$$ 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 $$ {Hol}(s_0)$$ is factorial. This holds also if $$s_0$$ is a zero of an irreducible L-function of dimension $$\le 2$$ , without any condition on the Galois group.