Compositio Mathematica

On the Exceptional Zeros of Rankin–Selberg L-Functions

Dinakar Ramakrishnan a1 and Song Wang a2
a1 1253-37 Caltech, Pasadena, CA 91125, U.S.A. e-mail: [email protected]
a2 School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, U.S.A. e-mail: [email protected]

Article author query
ramakrishnan d   [Google Scholar] 
wang s   [Google Scholar] 


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).

Key Words: 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.