On Non-Vanishing of Convolution of Dirichlet Series

We study the non-vanishing on the line Re(s) = 1 of the convolution series associated to two Dirichlet series in a certain class of Dirichlet series. The non-vanishing of various L-functions on the line Re(s) = 1 will be simple corollaries of our general theorems. Let f(z) = ∑∞ n=1 âf (n)e 2πinz and g(z) = ∑∞ n=1 âg(n)e 2πinz be cusp forms of weight k and level N with trivial character. Let Lf (s) = ∑∞ n=1 af (n)n −s and Lg(s) = ∑∞ n=1 ag(n)n −s be the L-functions associated to f and g, respectively, where af (n) = âf (n)/n k−1 2 and ag(n) = âg(n)/n k−1 2 . Let L(f ⊗ g, s) = ζN (2s) ∞ ∑ n=1 af (n)ag(n) ns be the Rankin-Selberg convolution of Lf (s) and Lg(s). In [11] Rankin established the analytic continuation of L(f⊗g, s) (see Theorem 1.5). Rankin’s Theorem has numerous number theoretic applications. In [10], Rankin used this theorem to prove the non-vanishing of the modular Lfunction associated to the discriminant function ∆(z) = e ∞ ∏