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On the Exceptional Zeros of Rankin–Selberg L-Functions

Published online by Cambridge University Press:  04 December 2007

Dinakar Ramakrishnan  and
Song Wang
Dinakar Ramakrishnan
Affiliation:
1253-37 Caltech, Pasadena, CA 91125, U.S.A. e-mail: dinakar@its.caltech.edu
Song Wang
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, U.S.A. e-mail: swang@ias.edu
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Abstract

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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 on the norm of sym2(g).

Keywords

factorizationsGL(n)holomorphic formsLandau–Siegel zeroslower boundMaass formsPetersson normpositive Dirichlet seriesRankin–Selberg L-functionsselfdual formsspectral normalizationsymmetric power liftingssymmetric space
Type
Research Article
Information
Compositio Mathematica , Volume 135 , Issue 2 , January 2003 , pp. 211 - 244
Copyright
© 2003 Kluwer Academic Publishers