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Is there an L-function ($L_s=\sum_{n =1}^{\infty} \frac{a_n}{n^s}$) having a functional equation coming from a relation of the type [1]: $\sum_{n =1}^{\infty} a_n \; e^{-2\pi nx}= \frac{A}{x^k} \sum_{n =1}^{\infty} a_n'\; e^{-2\pi\frac{n}{Nx}}$ but not satisfying the Riemann Hypothesis?

Notice that in the above sum there is no term for $n=0$.

(I think Dirichlet L-function and some Modular forms L-function with Fricke involution satisfy such relation and RH, but is this possibly a sufficient condition? Any counter example?)

The L-fonction I know having a functional equation and satisfying RH have also an associated relation of the type [1] (coming from Poisson Summation formula or Fricke involution) sometimes they have a term in 0 (like for Zeta) sometimes not (like for Dirichlet L-functions). Now I know example of L-functions not satisfying RH with functional equation but it seems they appear only with combination of L-functions having a term n=0 in the Poisson summation like formula.

By the way do you know a L-function satisfying RH without a relation like [1] (with possibly a term n=0)?

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Are you assuming the $a_n$'s are multiplicative? –  Micah Milinovich Aug 15 '13 at 16:31
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If you take two cuspidal eigenforms of the same weight and level, any linear combination of them will satisfy your functional equation, and have no term at zero. But it is easy to construct such a combination with a zero not on the critical strip. Like quid said, the associated L-function won't have an Euler product. I think we really expect only eigenforms to satisfy RH. –  Dror Speiser Aug 16 '13 at 7:01

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There certainly are L-functions having a functional equation and violating the analogue of RH. (I am not completely sure I understand what type of functional equation you mean by your restriction, but the ones I mention satisfy a functional equation of so to say the usual form, so I hope this is fine.)

Eppstein zeta-functions are a classical example; Davenport and Heilbronn 'On the zeros of certain Dirichlet series' (JLMS) is a classical reference.

For recent investigations in such phenomena see for example Frank Thorne's Analytic properties of Shintani zeta functions.

You might also have a look at the closely related MO question Are there refuted analogues of the Riemann hypothesis? where some information related to this is to be found.

In particular, it seems that for RH one needs Euler product also (not just a functional equation) and then there might be an equivalent; see Frank Thorne's answer on the above mentioned question for more specific information.

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Thanks for your answer. I did read the post "Are there refuted analogues of the Riemann hypothesis?" and I did read also the recommended "Zeros of the Davenport-Heilbronn Counterexample" but there the L-functions not satisfying RH are combination of Zeta and Dirichlet L-function. So the resulting L-function satisfy a relation looking like the type I mention above (due to Poisson summation formula) but there is a term for n=0 in this relation. I add explanation to my question. –  Bertrand Aug 16 '13 at 6:27
    
I am sorry I did not fully understand the point of your question right away. Thank you for the clarification. I am nout sure I can add anything beyond what is said already. Perhaps Dror Speiser's comment resolves the problem. –  quid Aug 16 '13 at 9:34

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