Zeros and poles of Artin <i>L</i>-series

Abstract

Let E/F be a finite normal extension of number fields with Galois group G . For each virtual character χ of G , denote by L ( s , χ) = L ( s , χ, F ) the Artin L -series attached to χ. It is defined for Re ( s ) &gt; 1 by an Euler product which is absolutely convergent, making it holomorphic in this half plane. Artin's holomorphy conjecture asserts that, if χ is a character, L ( s , χ) has a continuation to the entire s -plane, analytic except possibly for-a pole at s = 1 of multiplicity equal to 〈χ, 1〉, where 1 denotes the trivial character. A well-known group-theoretic result of Brauer implies that L ( s , χ) has a meromorphic continuation for all s .