On the Poles of Rankin-selberg Convolutions of Modular Forms

The Rankin-Selberg convolution is usually normalized by the multiplication of a zeta factor. One naturally expects that the non-normalized convolution will have poles where the zeta factor has zeros, and that these poles will have the same order as the zeros of the zeta factor. However, this will only happen if the normalized convolution does not vanish at the zeros of the zeta factor. In this paper, we prove that given any point inside the critical strip, which is not equal to 1 2 and is not a zero of the Riemann zeta function, there exist infinitely many cusp forms whose normalized convolutions do not vanish at that point. Introduction Assume that k is a positive even integer. Let Γ be the modular group SL2(Z). Denote by θk the dimension of the space Sk(Γ) of cusp forms. Assume that functions fjk(z) = ∞ ∑ n=1 ψjk(n)e(nz), j = 1, 2, · · · , θk, form an orthogonal base of Hecke eigenforms of Sk(Γ) such that the norm of fjk(z) is one under the Petersson inner product of the space. Put