On a mean value result for a product of $L$-functions

The asymptotic behaviour of moments of $L$-functions is of special interest to number theorists and there are conjectures that predict the shape of the moments for families of $L$-functions of a given symmetry type. However, only some results for the first few moments are known. In this talk we will consider the asymptotic behaviour of the first moment of the product of a Hecke $L$-function and a symmetric square L-function. This is joint work with O. Balkanova, G. Bhowmik, D. Frolenkov.