Abstract
In this paper, some asymptotic formula is proved for the harmonic mollified second moment of a family of Rankin-Selberg L-functions. The main contribution is a substancial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average. Consequences : · new subconvexity bound, · exponential decay of the analytic rank, · non-vanishing result around the real axis.