On L-values of elliptic curves twisted by cubic Dirichlet characters

Given an elliptic curve over Q of analytic rank zero, its L-function can be twisted by an even primitive Dirichlet character of order , and in many cases its associated special L-value (, ) is known to be integral after normalising by certain periods.This paper determines the precise value of (, ) in terms of Birch-Swinnerton-Dyer invariants when = 3, classifies their asymptotic densities modulo 3 by fixing and varying , and presents a lower bound on the 3-adic valuation of (, 1), all of which arise from a congruence of modular symbols.These results also explain some phenomena observed by Dokchitser-Evans-Wiersema and by Kisilevsky-Nam.