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On the structure of the Selberg class. II: Invariants and conjectures. (English) Zbl 0982.11051
The authors continue (the first paper was published in [Acta Math. 182, 207-241 (1999; Zbl 1126.11335)]) their interesting research on the well-known axiomatic Selberg class $$\mathcal S$$ of Dirichlet series. Let $$\mathcal S^{\#}$$ denote a larger class where Ramanujan’s hypothesis and Euler’s product are dropped. A $$\gamma$$-factor of a function $$F\in\mathcal S^{\#}$$ is an expression of the form $$\gamma(s)=Q^s \prod_{j=1}^r \Gamma(\lambda_j s+\mu_j)$$, with $$Q>0$$, $$\lambda_j>0$$ and $$\operatorname{Re} \mu_j\geq 0$$ for $$j=1,...,r$$, such that $$\Phi(s)=\gamma(s)F(s)$$ satisfies a functional equation of the type $$\Phi(s)= \omega \bar{\Phi}(1-s)$$ where $$|\omega|=1$$. It is known that the functional equation satisfied by a function $$F\in\mathcal S^{\#}$$ with degree $$d_F>0$$ is not unique in the sense that the data $$Q$$, $$r$$, $$\lambda_j$$, $$\mu_j$$ and $$\omega$$ are not uniquely determined by $$F(s)$$.
In Theorem 1, the authors give an explicit transformation algorithm for any two $$\gamma$$-factors of $$F$$. For Dirichlet $$L$$-functions, all $$\gamma$$-factors are given in Proposition 2.1.
Then the concept of an invariant of $$\mathcal S^{\#}$$ is defined. Roughly speaking, the invariant is an expression depending on the data $$Q$$, $$r$$, $$\lambda_j$$, $$\mu_j$$, $$\omega$$ which remains stable for any data corresponding to the same function $$F$$. Theorem 1 enables the invariants of functions $$F\in\mathcal S^{\#}$$ to be characterized, which is done in Theorem 2.
After that the authors propose several new and known conjectures that illustrate the relevance of such invariants to the structure of the Selberg class $$\mathcal S$$. The last two theorems are devoted to relationships between these conjectures.

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
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##### References:
 [1] Conrey J. B., Duke Math. J. 72 pp 673– (1993) [2] Kaczorowski J., Acta Math. 182 pp 207– (1999) [3] M. R. Murty, Selberg’s conjectures and Artin L-functions, Bull. A. M. S. 31 (1994), 1-14. · Zbl 0805.11062 [4] A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Proc. Amal Conf. Analytic Number Theory, ed. by E. Bombieri et al., UniversitaA di Salerno (1992), 367-385; Collected Papers, vol. II, Springer Verlag (1991), 47-63. [5] Springer Lect. Notes Math. 627 pp 79– (1977)
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