Distribution of values of $L$-functions arising from a Riemann-type functional equation

We consider $L$-functions satisfying a Riemann-type functional equation at special points of the "completing" factor in the functional equation. There is no strict definition of what should be regarded as a «Riemann-type» functional but we put it simply as a functional equation of the form $F(s)=\overline{F(1-\overline{s})}$, where $F(s)$ is the "completed" $L$-function.

In this context, we take into account the notion introduced by Atle Selberg, called the Selberg class, but dropping a few unnecessary conditions to obtain the so-called extended Selberg class, although it is possible to use a more general concept.

In this work, we consider the distribution of non-zero $a$-points of the «completing» factor in the functional equation which essentially arises from the Euler gamma function. We study the behavior of the corresponding $L$-function at those points and introduce the notion of weak Gram law for this $L$-function under this setting.

This is a joint work with Jörn Steuding (University of Würzburg) and Athanasios Sourmelidis (TU Graz), and this generalizes our previous works with respect to the Riemann zeta function and Dirichlet series with periodic coefficients, including the works of Jörn Steuding, Justas Kalpokas and Maxim Korolev (Steklov Mathematical Institute).

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