Critical Values of Derivatives of Twisted Elliptic $L$-Functions

Abstract

Let $L(E/\Q,s)$ be the $L$-function of an elliptic curve $E$ defined over the rational field $\Q$. We examine special values of the derivatives $L^{\prime}(E,1,\chi)$ of twists by Dirichlet characters of $L(E/\Q,s)$ when $L(E,1,\chi)=0$.