Given an L-function of an elliptic curve of conductor $N$, the symmetric $n$-th power L-function is defined by the Euler product $L(s,E,\text{sym}^n)=\prod_{p\nmid N} \prod_{j=0}^n \left(1 - \frac{\alpha_p^{j} \beta^{n-j}_p}{p^s} \right)^{-1} \times \prod_{p|N} L_p(s)$