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.