It is conjecture that under certain conditions a L-function satisfies RH. Among these conditions there is the necessity for the L-function to have an Euler product. (Some L-functions with a functional equation but without Euler product are known to have non trivial zeros with real part between 1/2 and 1).

So the Euler product seems to be an essential ingredient to RH but what are the main properties involved by an Euler product for a L-function ? (For exemple specific bound known linked to Euler product ? or new relation for the L-function?) To my knowledge the results are very poor.

Note: The equality between L-Function and Euler product holds out of the critical strip but the Euler products directly constraints the non critical zeros to be on the critical line... so I wonder what could be the firts steps of a bridge linking Euler product and RH.

provedonly to the right of the critical strip (including the right edge of the critical strip in some cases, e.g., $L$-functions of nontrivial Dirichlet characters). Nobody claims the Euler product should be invalid everywhere in the critical strip, and actually in some cases it should be valid to the right of the critical line, but it is hopeless to prove anything like that at present. $\endgroup$ – KConrad Apr 17 '17 at 17:04