Twists by Dirichlet characters and polynomial Euler products of \(L\)-functions. II (Q6606731)

The authors continue investigating \(L\)-functions belonging to the Selberg class \(\mathcal{S}\). More precisely, they solve the question of the relation between the function's degree, conductor, and the existence of a polynomial Euler product.\N\NThe main result of the present paper is as follows. Suppose that the function \(F\) belongs to the class \(\mathcal{S}\) and has a degree 2, and its conductor \(q_F\) is square-free. Then it is shown that if \(F\) is weakly twist-regular at all primes \(p \not | q_F\), then \(F\) has a polynomial Euler product.\N\NNote that the first result of such kind inquires for the function \(F\) to be weakly twist-regular at all primes \(p \not = q_F\) was obtained by the authors in Part I [J. Number Theory 253, 1--16 (2023; Zbl 1536.11137)]. However, for the results in this paper, the authors cannot use the same method as in the aforementioned paper, and the main tool used here is a certain transformation formula for linear twists of \(L\)-functions from the extended Selberg class \({\mathcal{S}}^\#\).\N\NFor the entire collection see [Zbl 1530.11003].