The Dirichlet series of the Hasse–Weil L-function - MathOverflow most recent 30 from http://mathoverflow.net2013-05-18T11:13:16Zhttp://mathoverflow.net/feeds/question/111391http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/111391/the-dirichlet-series-of-the-hasseweil-l-functionThe Dirichlet series of the Hasse–Weil L-functionRH2012-11-03T16:30:12Z2012-11-03T19:36:14Z
<p>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.</p>
http://mathoverflow.net/questions/111391/the-dirichlet-series-of-the-hasseweil-l-function/111395#111395Answer by GH for The Dirichlet series of the Hasse–Weil L-functionGH2012-11-03T17:55:59Z2012-11-03T17:55:59Z<p>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).</p>