On the Poles of Rankin-Selberg Convolutions of Modular Forms

Li, Xian-jin

doi:10.1090/S0002-9947-96-01540-1

On the Poles of Rankin-Selberg Convolutions of Modular Forms
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by Xian-jin Li PDF

Trans. Amer. Math. Soc. 348 (1996), 1213-1234 Request permission

Abstract:

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 $\frac {1}{2}$ 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.

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