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On the representation of $$H$$-invariants in the Selberg class. (English) Zbl 1312.11073
From the introduction: The extended Selberg class of functions $$\mathcal S^{\sharp}$$, introduced by J. Kaczorowski and A. Perelli in [Acta Math. 182, No. 2, 207–241 (1999; Zbl 1126.11335)], is a general class of Dirichlet series $$F$$ satisfying certain conditions.
The smallest integer $$m \geq 0$$ such that $$(s-1)^mF (s)$$ is entire is denoted by $$m_F$$ and called the polar order of $$F$$ . It is easy to see (due to the functional equation and the Stirling formula for the gamma function) that the function $$(s -1)^{m_F} F (s)$$ is actually an entire function of order one.
In [Acta Arith. 104, No. 2, 97–116 (2002; Zbl 0996.11053)], J. Kaczorowski and A. Perelli obtained an interpretation of $$H$$-invariants and conductor as coefficients in a certain asymptotic expansion of the gamma factor of the functional equation and raised the problem of interpreting $$H_F (n)$$, $$n \geq 2$$, in terms of $$F$$ alone, without explicit reference to the functional equation. The purpose of this paper is to give a solution of this problem.

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
##### Keywords:
$$H$$-invariants; Selberg class; zeta regularized products
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