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 $F(s,alpha)$.
Analytic properties of the standard twist of L-functions
A. Perelli
2021-01-01
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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