Let $E:y^2＝x^3-x/ \Bbb{Q}(i)$ be elliptic curve and $L(E,1)$ be a special value of $L$ function of $E$ at $1$.


Let $L(ψ,1)$ be value at $1$ of Hecke $L$ function with respect to Hecke character $ψ$, It is known that $L(E,1)＝L( \bar{ψ},1)L(ψ,1)$.


In this case, why $L(ψ,1)＝1$ ?


I may forget some trivial fact about Hecke $L$ character.
Thank you for your help.



Special value of Hecke $L$ function
Not clear what you mean by $\varphi$ or $\psi$. At any rate, $E$ is a CM curve, hence $L(s,E)$ equals $L(s,\psi)$ for a suitable Hecke character $\psi$ of the number field $\mathbb{Q}(i)$. You can find the details (e.g. the definition of $\psi$) in Sections 8.3-8.4 of Iwaniec's book "Topics in classical automorphic forms".


**Added.** As David Loeffler pointed out, if $E$ is regarded as a curve over $\mathbb{Q}(i)$, its $L$-function is not $L(s,\psi)$ but $L(s,\psi)L(s,\bar\psi)$. In my answer, $E$ was regarded as a curve over $\mathbb{Q}$, as was in the original question (cf. [revision history](https://mathoverflow.net/posts/426910/revisions)).


