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Let $E$ be an elliptic curve over $\mathbb{Q}$ with conductor $N$. Let $f=\sum a_n q^n$ be the weight 2 modular form corresponding to $E$. Define $L_2(f,s)=\zeta(s-1)L(Sym^2(E),s)$. The following identity is mentioned in Zagier's paper "Classical and alliptic polylogarithms and special values of L-series":

$L_2(f,s)=(1+N^{1-s})\zeta(2s-2)\sum_{n=1}^{\infty}\frac{a_n^2}{n^s}$.

I would like to find a proof of this identity. Thanks

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    $\begingroup$ Finally found a reference. See page 43 of Number theory 3 by Kurokawa,Kurihara and Saito (Translations of mathematical monographs Volume 242). The factor $(1+N^{1-s})$ comes from the primes of bad reduction and works for the case $N=37$. For different conductor, the formula may look different at the bad primes. $\endgroup$
    – Dianbin Bao
    Commented Jul 28, 2020 at 14:41
  • $\begingroup$ In (0.2) of "On the Holomorphy of Certain Dirichlet Series", Shimura says that "it can easily be seen that"... (though he does give the relevant Euler factor for the proof). $\endgroup$
    – OmniaOperator
    Commented Jul 28, 2020 at 16:23

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