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Weil’s converse theorem for Maass forms and cancellation of zeros

Volume 196 / 2020

Michael Neururer, Thomas Oliver Acta Arithmetica 196 (2020), 387-422 MSC: 11F66, 11M41, 11F12. DOI: 10.4064/aa190811-3-2 Published online: 11 July 2020

Abstract

We prove two principal results. Firstly, we characterise Maass forms in terms of functional equations for Dirichlet series twisted by primitive characters. The key point is that the twists are allowed to be meromorphic. This weakened analytic assumption applies in the context of our second theorem, which shows that the quotient of the symmetric square $L$-function of a Maass newform and the Riemann zeta function has infinitely many poles.

Authors

  • Michael NeururerFachbereich Mathematik
    Technische Universität Darmstadt
    Schloßgartenstr. 7
    64289 Darmstadt, Germany
    e-mail
  • Thomas OliverMathematical Institute
    Andrew Wiles Building
    University of Oxford
    Radcliffe Observatory Quater
    Woodstock Road
    Oxford, OX2 6GG, UK
    and
    Heilbronn Institute for Mathematical Research
    Bristol, UK
    e-mail

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