A hybrid asymptotic formula for the second moment of Rankin-Selberg L-functions

Blomer, Valentin and Harcos, Gergely (2012) A hybrid asymptotic formula for the second moment of Rankin-Selberg L-functions. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 105 (3). pp. 475-505. ISSN 0024-6115

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Abstract

Let g be a fixed modular form of full level, and let f j, k be a basis of holomorphic cuspidal newforms of even weight k, fixed level and fixed primitive nebentypus. We consider the Rankin-Selberg L-functions L(+it, fj, k ⊗ g) and compute their second moment over t T and k K. For K 3/4+epsi≤ T ≤ K 5/4-epsi, we obtain an asymptotic formula with a power-saving error term. Our result covers the second moment of L(+it+ir, fj, k)L(+it-ir, fj, k) for any fixed real number r, hence also the fourth moment of L(+it, fj, k). For the proof, we develop a precise uniform approximate functional equation with explicit dependence on the archimedean parameters. © 2012 London Mathematical Society.

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