Simultaneous non-vanishing of twists

Akbary, Amir

doi:10.1090/S0002-9939-06-08369-9

Simultaneous non-vanishing of twists
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Proc. Amer. Math. Soc. 134 (2006), 3143-3151 Request permission

Abstract:

Let $f$ be a newform of even weight $k$, level $M$ and character $\psi$ and let $g$ be a newform of even weight $l$, level $N$ and character $\eta$. We give a generalization of a theorem of Elliott, regarding the average values of Dirichlet $L$-functions, in the context of twisted modular $L$-functions associated to $f$ and $g$. Using this result, we find a lower bound in terms of $Q$ for the number of primitive Dirichlet characters modulo prime $q\leq Q$ whose twisted product $L$-functions $L_{f,\chi }(s_0) L_{g,\chi }(s_0)$ are non-vanishing at a fixed point $s_0=\sigma _0+it_0$ with $\frac {1}{2}<\sigma _0\leq 1$.

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