Simultaneous nonvanishing of automorphic L-functions at the central point

Let g be a holomorphic Hecke eigenform and {u _j } an orthonormal basis of even Hecke?CMaass forms for ${\textup{SL}(2,\mathbb{Z})}$ . Denote L(s, g × u _j ) and L(s, u _j ) the corresponding L-functions. In this paper, we give an asymptotic formula for the average of ${L(\frac{1}{2},g\times u_j)L(\frac{1}{2},u_j)}$ , from which we derive that there are infinitely many u _j ??s such that ${L(\frac{1}{2},g\times u_j)L(\frac{1}{2},u_j)\neq0}$ .