Average values of quadratic twists of modular $L$-functions

Abstract

This paper studies non-vanishing of quadratic twists of automorphic forms $f$ on $GL(2)$ over $\mathrm{Q}$ at various points inside the critical strip. Given any point $w_0$ inside the critical strip, and $\varepsilon > 0$, we show that at least $Y^{12/17-\varepsilon}$ of the quadratic twists $L(f,\chi_{d}, s)$ with $\vert d \vert \leq Y$ do not vanish inside the disc $\vert w - w_{0}\vert \lt (\mathrm{log}Y)^{-1-\varepsilon}$ (Here $d \equiv 1$ mod $4$ is a fundamental discriminant and $\chi_{d}$ denotes the Kronecker symbol.) If we assume the Ramanujan conjecture about the Fourier coefficients of $f$ (in particular, if $f$ is holomorphic) then $\frac{12}{17}$ above can be replaced with 1.

This should be compared with a result of Ono and Skinner [10] which states that if $f$ is a holomorphic newform of even weight and trivial character, then at least $\gg Y / \mathrm{log} Y$ of the quadratic twists $L(f,\chi_{d}, s)$ are nonzero at the central critical point. A slightly weaker result had been proved earlier by Perelli and Pomykala [11]. By contrast, we make no restriction on the holomorphy of $f$ and the result holds even if $f$ has non-trivial central character. Moreover, we prove non-vanishing in a disc about any point in the critical strip. As in [l1], our tools are the method of Iwaniec [4] and a mean value estimate of Heath-Brown [3].