On the Exceptional Zeros of Rankin–Selberg L-Functions |
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Authors: | Dinakar Ramakrishnan Song Wang |
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Institution: | (1) 1253-37 Caltech, Pasadena, CA, 91125, U.S.A.;(2) School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, U.S.A. |
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Abstract: | The main objects of study in this article are two classes of Rankin–Selberg L-functions, namely L(s,f×g) and L(s, sym2(g)× sym2(g)), where f,g are newforms, holomorphic or of Maass type, on the upper half plane, and sym2(g) denotes the symmetric square lift of g to GL(3). We prove that in general, i.e., when these L-functions are not divisible by L-functions of quadratic characters (such divisibility happening rarely), they do not admit any LandauSiegel zeros. Such zeros, which are real and close to s=1, are highly mysterious and are not expected to occur. There are corollaries of our result, one of them being a strong lower bound for special value at s=1, which is of interest both geometrically and analytically. One also gets this way a good bound onthe norm of sym2(g). |
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Keywords: | factorizations GL(n) holomorphic forms Landau– Siegel zeros lower bound Maass forms Petersson norm positive Dirichlet series Rankin– Selberg L-functions selfdual forms spectral normalization symmetric power liftings symmetric space |
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