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# Non-vanishing of $L$-functions, the Ramanujan Conjecture, and Families of Hecke Characters

Published:2011-12-23
Printed: Feb 2013
• Valentin Blomer,
Universität Göttingen, Mathematisches Institut, Bunsenstr. 3-5, 37073 Göttingen
• Farrell Brumley,
Institut Élie Cartan, Université Henri Poincaré Nancy 1, BP 239, 54506 Vandėuvre Cedex, France
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## Abstract

We prove a non-vanishing result for families of $\operatorname{GL}_n\times\operatorname{GL}_n$ Rankin-Selberg $L$-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on $\operatorname{GL}_n$. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.
 Keywords: non-vanishing, automorphic forms, Hecke characters, Ramanujan conjecture
 MSC Classifications: 11F70 - Representation-theoretic methods; automorphic representations over local and global fields 11M41 - Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

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