The L-function $L(s,E)= \sum a_n n^{-s}$ of an elliptic curve of conductor $N$ has an Euler product of the form \[ L(s,E)=\prod_{p\mid N} \left(1 - a_p p^{-s} \right)^{-1}\prod_{p\nmid N} \left(1 - a_p p^{-s} + p^{-2s} \right)^{-1} \] and satisfies the functional equation \[ \Lambda(s,E)= N^{s/2}\Gamma_{\mathbb C}(s+1/2)\cdot L(s,E)= \varepsilon\Lambda(1-s,E), \] where the sign \(\varepsilon\) is equal to either \(+1\) or \(-1\).