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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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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