Non-vanishing of Artin-twisted L-functions of Elliptic Curves

Abstract

Let E be an elliptic curve and \rho an Artin representation, both defined over the rational numbers. Let p be a prime at which E has good reduction. We prove that there exists an infinite set of Dirichlet characters \chi, ramified only at p, such that the Artin-twisted L-values L(E,\rho \chi,\beta) are non-zero when \beta lies in a specified region in the critical strip (assuming the conjectural continuations and functional equations for these L-functions). The new contribution of our paper is that we may choose our characters to be ramified only at one prime, which may divide the conductor of \rho.