Vanishing and non-vanishing of L-series of elliptic curves twisted by Dirichlet characters

We study the behaviour of L-series of elliptic curves twisted by Dirichlet characters. In particular, we study the vanishing and non vanishing of these L-series at the critical point. We present empirical results indicating the vanishing behaviour of cyclic twists of orders 3, 5, 7 and conductors up to 5000 for elliptic curves of conductor less than 100. We prove results for vanishing in the case of cyclic cubic twists and non-vanishing in the case of cyclic twists of arbitrary prime order.; Let L(E, s) be the L-series of an elliptic curve E : y2 = x3 + Ax + B with A, B ∈ $Q$.; If there exists a cyclic cubic character χ such that L( E, 1, χ) = 0 or if L (E, 1) = 0 then the L-series vanishes for an infinite number of cyclic cubic characters.; With finite exceptions, if L(E, 1) ≠ 0 there exist an infinite number of cyclic twists $y$ of prime order k such that L( E, 1, $y$) ≠ 0 for every order k.