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I know that if you have an elliptic curve E with complex multiplication, then the Hasse Weil L-series attached to it can be expressed in terms of Hecke L-series. Is there anything that can be said when the curve doesn't necessarily have CM or any reason why a similar fact would not hold?

On a related note, any basic facts/discussions of elliptic curves or Hecke L-series that you think is interesting or good to know would be greatly appreciated!

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Yes, there are some theorems, but they are a little bit harder. Look on wikipedia: – KConrad Jul 2 '10 at 0:06

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