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)?