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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$\begingroup$ I added the reference-request tag since this question asking for just that. $\endgroup$– BSteinhurstCommented Nov 3, 2012 at 16:35
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$\begingroup$ What makes a dirchlet series finite order? $\endgroup$– Will SawinCommented Nov 3, 2012 at 17:12
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$\begingroup$ Please see this link to clarify the idea: math.harvard.edu/~elkies/M259.06/prod.pdf $\endgroup$– SafwaneCommented Nov 3, 2012 at 17:20
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3$\begingroup$ 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. $\endgroup$– user9072Commented Nov 3, 2012 at 19:06
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1 Answer
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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).
answered Nov 3, 2012 at 17:55