  We obtain a first moment formula for Rankin-Selberg convolution $L$-series of
holomorphic modular forms or Maass forms of arbitrary level on $GL(2)$, with an
orthonormal basis of Maass forms. One consequence is the best result to date,
uniform in level, spectral value and weight, for the equality of two Maass or
holomorphic cusp forms if their Rankin-Selberg convolutions with the
orthonormal basis of Mass forms $u_j$ is equal at the center of the critical
strip for sufficiently many $u_j$.
  The main novelty of our approach is the new way the error terms are treated.
They are brought into an exact form that provides optimal estimates for the
first moment case, and also provide a basis for an extension to second moments,
which will appear in another work.

Number Theory