Hecke L-function ($L(\psi ,s)={\prod }_{\mathfrak{q}\nmid \mathfrak{f}}(1-\psi (\mathfrak{q})N{\mathfrak{q}}^{-s}{)}^{-1}$)

The L-function of a Hecke character is initially the displayed Euler product over prime ideals away from its conductor. The equivalent Dirichlet series sums $\psi (\mathfrak{a})N{\mathfrak{a}}^{-s}$ over integral ideals prime to the conductor. For a CM elliptic curve with its endomorphisms defined over the base, its Hasse-Weil L-function factors into the two conjugate Hecke L-functions.