Artin’s conjecture on entireness of Artin L-functions for nontrivial irreducible representations

Prove that for any nontrivial irreducible complex Galois representation ρ of the Galois group of a number field extension, the associated Artin L-function L(O_F, ρ, s) extends to an entire function on the complex plane.

Background

The authors review analytic properties of Artin L-functions, noting Artin’s conjecture that entireness should hold for all nontrivial irreducible representations, and that the conjecture is known in dimension one via Artin reciprocity (abelian case).

The conjecture is central to the analytic theory of L-functions and underlies much of the motivic and cohomological structure connected to the values of L-functions, including results discussed in the paper.