  In this paper, some asymptotic formulas are proved for the harmonic mollified
second moment of a family of Rankin-Selberg L-functions. One of the main new
input is a substantial improvement of the admissible length of the mollifier
which is done by solving a shifted convolution problem by a spectral method on
average. A first consequence is a new subconvexity bound for Rankin-Selberg
L-functions in the level aspect. Moreover, infinitely many Rankin-Selberg
L-functions having at most eight non-trivial real zeros are produced and some
new non-trivial estimates for the analytic rank of the family studied are
obtained.
