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)? 



Sufficient condition for Riemann Hypothesis?
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](http://www.math.sc.edu/~thornef/shintani-proceedings-final.pdf). 


You might also have a look at the closely related MO question [Are there refuted analogues of the Riemann hypothesis?](https://mathoverflow.net/questions/127520/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.


