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ON THE STANDARD TWIST OF THE $L$-FUNCTIONS OF HALF-INTEGRAL WEIGHT CUSP FORMS

Part of: Zeta and $L$-functions: analytic theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 December 2018

JERZY KACZOROWSKI  and
ALBERTO PERELLI
JERZY KACZOROWSKI
Affiliation:
Faculty of Mathematics and Computer Science, A.Mickiewicz University, 61-614 Poznań, Poland Institute of Mathematics of the Polish Academy of Sciences, 00-956 Warsaw, Poland email [email protected]
ALBERTO PERELLI
Affiliation:
Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy email [email protected]
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Abstract

The standard twist $F(s,\unicode[STIX]{x1D6FC})$ of $L$-functions $F(s)$ in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for example the functional equation. Here we deal with a special case, where $F(s)$ satisfies a functional equation with the same $\unicode[STIX]{x1D6E4}$-factor of the $L$-functions associated with the cusp forms of half-integral weight; for simplicity we present our results directly for such $L$-functions. We show that the standard twist $F(s,\unicode[STIX]{x1D6FC})$ satisfies a functional equation reflecting $s$ to $1-s$, whose shape is not far from a Riemann-type functional equation of degree 2 and may be regarded as a degree 2 analog of the Hurwitz–Lerch functional equation. We also deduce some results on the growth on vertical strips and on the distribution of zeros of $F(s,\unicode[STIX]{x1D6FC})$.

MSC classification

Primary: 11M41: Other Dirichlet series and zeta functions 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11F37: Forms of half-integer weight; nonholomorphic modular forms
Type
Article
Information
Nagoya Mathematical Journal , Volume 240 , December 2020 , pp. 150 - 180
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Copyright
© 2018 Foundation Nagoya Mathematical Journal

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