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 : y 2 = x 3 + Ax + B with A, B ✹ [Special characters omitted.] . If there exists a cyclic cubic character { such that L ( E , 1, {) = 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 [Special characters omitted.] of prime order k such that L ( E , 1, [Special characters omitted.] ) ✹ 0 for every order