Riv. Mat. Univ. Parma, Vol. 12, No. 1, 2021

Jerzy Kaczorowski [a,b] and Alberto Perelli [c]

Analytic properties of the standard twist of \(L\)-functions

Pages: 125-141
Received: 24 November 2019
Accepted in revised form: 15 April 2020
Mathematics Subject Classification (2010): 11M41.
Keywords: \(L\)-functions, standard twist, convexity bounds, distribution of zeros, nonlinear exponential sums, Selberg class.
Authors address:
[a]: Faculty of Mathematics and Computer Science, A. Mickiewicz University, 61-614 Poznań, Poland
[b]: Institute of Mathematics, Polish Academy of Sciences, 00-956 Warsaw, Poland
[c]: Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy

This research was partially supported by the Istituto Nazionale di Alta Matematica, by the MIUR grant PRIN-2015 ''Number Theory and Arithmetic Geometry'' and by grant 2017/25/B/ST1/00208 ''Analytic methods in number theory'' from the National Science Centre, Poland

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Abstract: We study some consequences of the functional equation satisfied by the standard twist \(F(s,\alpha)\) of the \(L\)-functions \(F(s)\) from the extended Selberg class. The shape of such a functional equation differs significantly from the classical one of Riemann-type satisfied by \(F(s)\); for example, it contains an error term which can identically vanish only in some special but well described cases. In this paper we show that this unusual functional equation can nevertheless be used to investigate convexity bounds, asymptotic formulae for the average of the coefficients and distribution of zeros of \(F(s,\alpha)\).

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