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.

References

Artin conjectured that for any non-trivial irreducible Galois representation $\rho$ of number fields, the Artin $L$-function $L(X,\rho, s)$ extends to an entire function of $s$.

— Equivariant algebraic $\mathrm{K}$-theory and Artin $L$-functions  (2405.03578 - Elmanto et al., 2024) in Remark, Section 2.2 (Artin L-functions of Galois representations)

The $L$-function has a meromorphic continuation to the whole complex plane and it is conjectured to be entire if $\rho$ is non-trivial.

— On multiply induced characters  (2609.04764 - Kida, 4 Sep 2026) in Section 2, Number-theoretic background and consequences