L

Abstract

. We prove a non-vanishing result of modular L -values with quadratic twists, where the quadratic discriminants are in a short interval. Us- ing this fact and Waldspurger’s theorem, we improve the results of Balog-Ono[The chebotarev density theorem in short intervals and some questions of Serre, Journal of number theory. 91(2):356-371(2001)] on the non-vanishing of Fourier coefficients of half-integral weight eigenform.