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Quantitative non-vanishing results on \(L\)-functions. (English) Zbl 1456.11169

Summary: In this paper, we establish a quantitative non-vanishing result on a class of twisted \(L\)-functions \(\mathcal{A}(s,\chi)\) of degree \(k\), which satisfy some mild and standard assumptions. As a corollary, we show that for a positive proportion of characters in a specific set, the special values \(\mathcal{A}(\beta,\chi)\) are non-vanishing for \(\operatorname{Re}\beta>1-\frac{1}{k}\). In particular, our argument holds for Rankin-Selberg \(L\)-functions and symmetric square \(L\)-functions on certain higher rank groups, and leads to some new non-trivial results for the first time.

MSC:

11M41 Other Dirichlet series and zeta functions
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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