Details

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

Structural invariants of \(L\)-functions and applications: a survey

Pages: 137-159
Received: 9 March 2021
Accepted in revised form: 8 April 2021
Mathematics Subject Classification: 11M41.
Keywords: Invariants of \(L\)-functions, standard twist, Selberg class, functional equation, modular forms.
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-2017 "Geometric, algebraic and analytic methods in arithmetic" and by grant 2017/25/B/ST1/00208 "Analytic methods in number theory" from the National Science Centre, Poland.

Abstract: This is a survey of the structural invariants of the \(L\)-functions in the extended Selberg class \(\mathcal S^\sharp\), covering some of their applications. In particular, we deal with the applications to the functional equation of the standard twist and to the classification of the functions in \(\mathcal S^\sharp\) of degree \(d=2\) and conductor \(q=1\). Moreover, we give a new, purely algebraic, definition of the structural invariants and provide an explicit expression for them.

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