  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.
