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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.

A p-adic L-function is a $p$-adic analytic function or measure interpolating suitably adjusted special values of an L-function. The Kubota-Leopoldt p-adic L-function is the basic example attached to a Dirichlet character.