Average values of quadratic twists of modular \(L\)-functions (Q5933540)

The \(L\)-functions \(L(f,\chi _d,s)\) under consideration are attached to a (holomorphic or non-holomorphic) cusp form \(f\) of given level, weight and character, which is a normalized eigenform for the Hecke operators, and its \(L\)-function is twisted by a primitive real character \(\chi _d\) \(\pmod{|d|}\). The main result states that given a point \(w _0\) inside the critical strip, there are at least \(Y^{12/17-\varepsilon}\) such \(L\)-functions with \(|d|\leq Y\) which do not vanish inside the disc \(|w-w_0|< (\log Y)^{-1-\varepsilon}\). If we assume the Ramanujan conjecture about the Fourier coefficients of \(f\) (in particular, if \(f\) is holomorphic), then \(12/17\) can be replaced by 1. The proof is based on mean value estimates for the twisted \(L\)-functions with respect to \(d\).