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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.

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I added the reference-request tag since this question asking for just that. – BSteinhurst Nov 3 '12 at 16:35
What makes a dirchlet series finite order? – Will Sawin Nov 3 '12 at 17:12
Please see this link to clarify the idea: – China-Hong Kong Nov 3 '12 at 17:20
I deleted an answer I gave, as it could create unnecessary complications. @Zeraoulia: could you please make it a habit to ask what you actually want. From the comment on my deleted answer it appears you in fact have a paper claiming the result you ask about. – user9072 Nov 3 '12 at 19:06
up vote 6 down vote accepted

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 Phragmén-Lindelöf convexity principle that it is of order 1 (and much more).

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