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The Selberg class is an axiomatization of arithmetically significant zeta functions (a.k.a. L-functions) by a few analytic properties (functional equation etc.) However there do exist other zeta functions that do not seem to come from arithmetic, but instead from geometry, for example. Some of these are known to have zeros on the real axis to the right of the critical line, but are otherwise expected to satisfy an analogue of the Riemann Hypothesis.

Have any attempt been made to write down axioms for such more general zeta functions that are expected to satisfy an appropriately modified Riemann Hypothesis?

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    $\begingroup$ @Marius: Can you give an example to the following: "Some of these are known to have zeros on the real axis to the right of the critical line, but are otherwise expected to satisfy an analogue of the Riemann Hypothesis." $\endgroup$
    – GH from MO
    Aug 9, 2011 at 14:42
  • $\begingroup$ @GH: Actually, I was quoting Atle Selberg from a long interview that he gave about a year before he died. I have to scratch around for the interview, but suspect that he was thinking of the Selberg zeta function for a generic compact Riemann surface. I will report back. $\endgroup$
    – Marius Overholt
    Aug 9, 2011 at 14:55
  • $\begingroup$ (There are also dynamical zeta functions: ams.org/notices/200208/fea-ruelle.pdf ) $\endgroup$
    – Qfwfq
    Aug 9, 2011 at 15:05
  • $\begingroup$ @GH: It was the way I thought I remembered it. Selberg made the remark in part 3 of the interview, about the Riemann Hypothesis and the trace formula. The interviewers were Nils A. Baas and Christian F. Skau. I translate their question from Norwegian into English: Have you considered whether there exists any kind of geometrical analogies to the primes in a fundamental sense? Selberg's answer translated from the Norwegian: $\endgroup$
    – Marius Overholt
    Aug 10, 2011 at 6:03
  • $\begingroup$ If you look at a compact Riemannian surface with the hyperbolic metric, and consider the closed geodesics, you can say that their lengths correspond to the logarithms of the primes. In the compact case one knows that the Riemann Hypothesis essentially holds, except that in some particular cases we have some zeros between 1/2 and 1 on the real axis, which I do not think can happen with those functions that we usually consider in number theory. Though I know that some have believed that there may be quadratic L-functions with zeros between 1/2 and 1. $\endgroup$
    – Marius Overholt
    Aug 10, 2011 at 6:11

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It is not true that the Selberg class includes only L-functions that "come from arithemtic". In fact, all the "reasonable geometric L-functions", that is, motivic L-functions are expected to belong to it.

For an unconditional example, the Hasse-Weil L-function of elliptic curves over $\mathbb{Q}$ are known to be in the Selberg class (ultimately this follows from the modularity theorem of Breuil-Conrad-Taylor-Wiles).

The only class of L-functions more general than Selberg's that (as far as I know) has been studied in some depth is the extended Selberg class $S♯$, which removes the Ramanujan and Euler product restrictions (see, for example Kaczorowski & Perelli (2002) On the structure of the Selberg class, V)

The problem with any further extension of the axioms is that it allows L-functions that behave too badly for any reasonable version of the Riemann hypothesis to hold.

In fact, the reason that makes extended Selberg class interesting is that anything that so far has been proved about Selberg class can be proved in the extended setting too, and restricted after that if necessary (Kaczorowski & Perelli classification results are directly on the extended case, for example).

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Andrew Booker defined the notion of L-datum that encompasses the Selberg class and allows to obtain simplicity results for the non trivial zeroes of some automorphic L-functions. Thomas Oliver, building on his work, began to classify L-data of low degrees.

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