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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 $L$-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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The Dirichlet series of the Hasse–Weil L-function

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.