Lower bounds for the conductor of \(L\)-functions (Q2919679)

In this paper, the authors give (as the title alludes to) lower bounds for the conductor of \(L\)-functions, namely in the three classes of such functions, all related to the famous Selberg class (SC), introduced by Selberg in his famous work in the Proceedings of the Amalfi conference (see [\textit{A. Selberg}, in: Proceedings of the Amalfi conference on analytic number theory, September, 1989. Salerno: Universitá di Salerno. 367--385 (1992; Zbl 0787.11037)]).NEWLINENEWLINEThe \textit{conductor} is a technical quantity, like the degree; they are defined in the SC but already in a larger class, ESC, see the following, as they are related to the functional equation.NEWLINENEWLINEThese are the Selberg slass \({\mathcal S}\), a class of Dirichlet series with good complex analytic behaviour, functional equation, Euler product and satisfying Ramanujan conjecture, described by five axioms; the extended Selberg class (ESC), more comprehensive, since doesn't require the two arithmetic axioms: Euler product and the Ramanujan conjecture and a subset of SC, consisting of those Dirichlet series in \({\mathcal S}\) which have a polynomial Euler product, denoted \({\mathcal S}^*\). These, actually are conjectured to be exactly all of SC (i.e., conjecturally \({\mathcal S}^*={\mathcal S}\)).NEWLINENEWLINEThe present paper is a study, in these three sets, relating the so-called \textit{invariants} of an \(L\)-function \(F\); namely those quantities, usually complex numbers, in a sense, characterizing \(F\). Examples of these are the degree, the root number, the conductor and the \(H\)-invariants, numbers \(H=H_F(n)\) depending on integers \(n\geq 0\) (involving \(n\)-th Bernoulli polynomials and the \(\Gamma\)-factors arguments in the functional equation). Actually, these four invariants are enough, for implying the same functional equation, in ESC.NEWLINENEWLINEHere the authors introduce in ESC, two new invariants coming from the \(H\)-numbers and one straight from the functional equation. These first two are related in Theorem 1. Then, these three new invariants are used in Theorems 2 and 3, to give lower bounds for the conductor in \({\mathcal S}^*\) (conjecturally, in SC), involving the degree, too.NEWLINENEWLINEJust to give an idea of the proofs, they are standard in analytic number theory, applying complex analysis, Hadamard product (from the axioms) and Turán's second main theorem.NEWLINENEWLINENotwithstanding the technical nature of the paper, the properties and all of the necessary links, to the literature, are clearly stated and proved in an elegant way.NEWLINENEWLINELast but not least, the authors ``collect further results and problems, and consider several special cases''.