Selberg defined what is now known as the Selberg class as a class of L-functions fulfilling for essential properties, which are analyticity, Euler product, functional equation and Ramanujan-Patersson condition. All elements of this class are believed to fulfill the analogue of the Riemann hypothesis, which translates in terms of distribution of the real parts of the non trivial zeros as a Dirac distribution centered in one half.

I would like to analyze the correspondence between properties defining the Selberg class and properties of the related distribution of the real parts of the non trivial zeros.

First, analyticity : as zeros of an analytic complex valued function are isolated, the distribution of real parts of the non trivial zeros (call it "the core distribution" of the considered L-function) is discrete.

Second, the Euler product : this implies the L-function doesn't vanish outside the critical strip if one puts aside the trivial zeros. Therefore the core distribution is compactly supported.

Third, the functional equation : this implies the core distribution is symmetric.

Finally the Ramanujan-Petersson condition : this one appears to be the most mysterious in terms of properties of the core distribution.

Thus my question: does it translate as strong unimodality thereof? If yes, does the conjunction of all those properties suffice to make the core distribution the Dirac distribution centered in one half?


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