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Let $E$ be an ellitpic curve over a quadratic field $K/\mathbb{Q}$. Then the L function of $E$ is defined as

$L(E_K,s)=\prod_{\mathfrak{p}\nmid \Delta}(1-a_{\mathfrak{p}}N(\mathfrak{p})^{-s}+N(\mathfrak{p})^{1-2s})\prod_{\mathfrak{p}|\Delta}(1-a_{\mathfrak{p}}N(\mathfrak{p})^{-s})$, where $a_{\mathfrak{p}}=N(\mathfrak{p})+1-\#E(\mathcal{O}_K/\mathfrak{p})$.

Given an elliptic curve explicitly, how to find effectively the Dirichlet series of $L(E_K,s)$ and its functional equation?

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    $\begingroup$ One can find the conductor with Tate's algorithm. Then look for Hilbert modular forms / Bianchi modular forms of the corresponding level. Or, if the conductor is small, look it up here lmfdb.org/EllipticCurve $\endgroup$
    – Will Sawin
    Commented Jul 26, 2020 at 18:19
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    $\begingroup$ Many computer algebra systems can do this for you. For instance in pari/gp, use nfinit (for $K$) + ellinit (for $E$) + lfuncreate (for $L$) (look up the documentation to learn how to use those). $\endgroup$
    – Aurel
    Commented Jul 26, 2020 at 19:28
  • $\begingroup$ Can you be a little more precise about what you mean? How to find the first $n$ coefficients of the Dirichlet series from the $a_p$'s, and the analytic conductor and the root number? In what sense do you have the elliptic curve and what sense do you want the L-function? $\endgroup$
    – Kimball
    Commented Jul 26, 2020 at 23:14
  • $\begingroup$ $a_{\mathfrak{p}}$ can be computed by counting. To me it seems not effective to compute $a_n$ by the Euler product. I want the dirichelt series of the L function and its functional equation to compute numerically the critial values of the L functions. By the comment of Will above, it seems to corresponds to Hilbert modular forms or Bianchi modulars depending whether the quadratic field is real or complex? $\endgroup$
    – Dianbin Bao
    Commented Jul 27, 2020 at 0:31
  • $\begingroup$ In practice, computing critical values is more or less equivalent to computing the first $O(\sqrt N)$ L-series coefficients. I'm not sure it will ever be more efficient to compute critical values by a correspondence to a Hilbert/Bianchi form. (In the case over Q, this only occurs when there is an easy eta-product for the modular form, or possibly when there is CM (both are easy to compute in that case)). OTOH, some theoretical constructs (ie period relations) might be easier for the form rather than the curve. $\endgroup$
    – OmniaOperator
    Commented Jul 27, 2020 at 2:41

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