On the vanishing of twisted L-functions of elliptic curves

Let E be an elliptic curve over Q with L-function L-E(s). We use the random matrix model of Katz and Sarnak to develop a heuristic for the frequency of vanishing of the twisted L-functions LE(I,chi), as chi runs over the Dirichlet characters of order 3 (cubic twists). We also compute explicitly the conjecture of Keating and Snaith about the moments of the special values LE(I,chi) in the family of cubic twists. Finally, we present experimental data which is consistent with the conjectures for the moments and for the vanishing in the family of cubic twists of L-E(s).