# Functions with and without Euler Product what is the difference ??

let be a function $f(s)$ which satisfy the Riemann functional equation

$f(1-s)=R(s)f(s)$

however there are cases that $f(s)$ satisfy Riemann Hypothesis for its Nontrivial zeros (assumed) but has an Euler product $f(s)= \prod _{p} \frac{1}{1-p^{-s}}$

in other cases $f(s)$ has zeros OUTSIDE the critical line but has no Euler product

if there a geometric or Number theoretic explanation or similar for an Euler product ?? apart from the fact that taking the logarithmic derivative inside a product can be interpreted in terms of closed orbits.

-
 The number-theoretic significance is explained at mathoverflow.net/questions/102799/… . – Qiaochu Yuan Jul 25 '12 at 13:54