We consider statistical properties of Hecke eigenvalues A j .p; 1/ for fixed p as j runs through a basis of Hecke-Maaß cusp forms for the group SL 3 .Z/. We show that almost all of them satisfy the Ramanujan conjecture at p and that their distribution is governed by the Sato-Tate law.
The goal of this paper is to derive a number theoretic expression for the trace tr l T v of the Hecke operator T v acting on the eigenspace E l for the eigenvalue l ¼ 1 þ k 2 > 0 of ÀD. We obtain that the trace can be expressed as the residue of a certain linear combination of L-series. Furthermore, the analytic behaviour of this combination gives information on the asymptotic behaviour of the class number.
In this paper various analytic techniques are combined in order to study the average of a product of a Hecke Lfunction and a symmetric square L-function at the central point in the weight aspect. The evaluation of the second main term relies on the theory of Maaß forms of half-integral weight and the Rankin-Selberg method. The error terms are bounded using the Liouville-Green approximation.2010 Mathematics Subject Classification. Primary: 11F12; Secondary: 33C05, 34E05, 34E20.
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