I have the following question: Is there is a paper claiming that the Dirichlet series of the Hasse–Weil $L$function (associated with an elliptic curve over rationals) is of finite order. Thank you in advance. I cannot find any result in the net.

Without the modularity theorem we don't even know that $L(E,s)$ is entire. The modularity theorem implies that $L(E,s)$ is the $L$function of a holomorphic cusp form (for a congruence subgroup), for which it follows by the functional equation and the PhragménLindelöf convexity principle that it is of order 1 (and much more). 

