The standard twist $F(s,alpha)$ 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 $Ga$-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 $F(s,alpha)$ 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 result on the growth on vertical strips and on the distribution of zeros of $F(s,alpha)$.
On the standard twist of the L-functions of half-integral weight cusp forms
PERELLI, A.
2020-01-01
Abstract
The standard twist $F(s,alpha)$ 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 $Ga$-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 $F(s,alpha)$ 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 result on the growth on vertical strips and on the distribution of zeros of $F(s,alpha)$.
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
