Classification of L-functions of degree 2 and conductor 1

An extension of Venkatesh’s converse theorem to the Selberg class

“…In this way, our result can be viewed as a converse theorem for degree elements of the Selberg class, albeit with infinitely many functional equations. Recently, Kaczorowski and Perelli [KP22] have classified the elements of the Selberg class of conductor without the need for any twists. Very little is known for higher conductor, however, and our result is the first that we are aware of to consider both arbitrary level and degree gamma factor. For , it is enough to assume the analytic properties (analytic continuation, finite order, functional equation) of the finite L -functions , and in this case we can also conclude that f is cuspidal.…”

Nonlinear twists and moments of $L$-functions

“…In our papers [1,[4][5][6][7] we studied the analytic properties of a class of nonlinear twists of F . Moreover, in these and other papers, notably in [8,9], we refined and applied such properties to the study of the structure of the Selberg classes.…”