In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, Theorem II.10.3, we have
Let $L(s,\psi)$ be the Hecke $L$-series attached to the Größencharakter $\psi$. Then $L(s,\psi)$ has an analytic continuation to the entire complex plane and satisfies a functional equation relating its values at $s$ and $N-s$ for some real number $N = N(\psi)$.
After reading Tate's thesis I'm somewhat uneasy about this claim. It seems that the functional equation should relate $L(s,\psi)$ to $L(1-s,\overline{\psi})$; and in particular, $N$ is always $1$.
The later results (II.10.5.1) seem to require a functional equation relating $L(s,\psi)$ to either $L(2-s,\overline{\psi})$ or $L(2-s,\psi)$. I don't know what this could be. Can anybody help?
Both statements are correct. What is going on here is that the Groessencharacter of an elliptic curve with CM satisfies some relation like $\overline{\psi(x)} = \psi (x^\sigma) \varepsilon(x)$, where $\varepsilon$ is the norm character and $x \mapsto x^\sigma$ is the action of complex conjugation on the ideles of $K$. Since multiplying a Groessencharacter by a power of $\varepsilon$ just shifts its Hecke L-series, this relates the functional equation a la Tate and a la Silverman.