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

Abstract

Let $L (E /\mathbb{Q} , s)$ be the $L$-function of an elliptic curve $E$ defined over the rational field $\mathbb{Q}$. Assuming the Birch–Swinnerton-Dyer conjectures, we examine special values of the $r$th derivatives, $L^{(r)}(E , 1, \chi)$, of twists by Dirichlet characters of $L (E /\mathbb{Q} , s)$ when $L (E , 1, \chi) = • • • = L^{(r−1)} (E , 1, \chi) = 0$.