A comparison of automorphic and Artin L-series of GL(2)-type agreeing at degree one primes

Abstract

Let F/k be a cyclic extension of number fields of prime degree. Let ρ be an irreducible 2-dimensional representation of Artin type of the absolute Galois group of F, and π a cuspidal automorphic representation of GL_2(A_F), such that the L-functions L(s,ρ_v) and L(s,π_v) agree at all (but finitely many of) the places v of degree one over k. We prove in this case that we have the global identity L(s,ρ)=L(s,π), with ρ_v↔π_v being given by the local Langlands correspondence at all v. In particular, π is tempered and L(s,ρ) is entire.

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