Poles of Artin $L$-functions and the strong Artin conjecture


We show that if the $L$-function of an irreducible 2-dimensional complex Galois representation over $\mathbb{Q}$ is not automorphic then it has infinitely many poles. In particular, the Artin conjecture for a single representation implies the corresponding strong Artin conjecture.


Andrew R. Booker

Department of Mathematics, Princeton University, Princeton, NJ 08544, United States