A Hecke character with modulus "m" is a group homomorphism from "I""m" into the nonzero complex numbers such that on ideals ("a") in "P""m" its value is equal to the value at "a" of a continuous homomorphism to the nonzero complex numbers from the product of the multiplicative groups of all Archimedean completions of "K" where each local component of the homomorphism has the same real part (in the exponent). (Here we embed "a" into the product of Archimedean completions of "K" using embeddings corresponding to the various Archimedean places on "K".) Thus a Hecke character may be defined on the ray class group modulo "m", which is the quotient "I""m"/"P""m". Strictly speaking, Hecke made the stipulation about behavior on principal ideals for those admitting a totally positive generator. So, in terms of the definition given above, he really only worked with moduli where all real places appeared. The role of the infinite part "m"∞ is now subsumed under the notion of an infinity-type. The ideal definition is much more complicated than the idelic one, and Hecke's motivation for his definition was to construct "L"-functions (sometimes referred to as Hecke "L"-functions) that extend the notion of a Dirichlet "L"-function from the rationals to other number fields. For a Hecke character χ, its "L"-function is defined to be the Dirichlet series  carried out over integral ideals relatively prime to the modulus "m" of the Hecke character. The notation "N(I)" means the ideal norm. The common real part condition governing the behavior of Hecke characters on the subgroups "P""m" implies these Dirichlet series are absolutely convergent in some right half-plane. Hecke proved these "L"-functions have a meromorphic continuation to the whole complex plane, being analytic except for a simple pole of order 1 at "s" = 1 when the character is trivial. For primitive Hecke characters (defined relative to a modulus in a similar manner to primitive Dirichlet characters), Hecke showed these "L"-functions satisfy a functional equation relating the values of the "L"-function of a character and the "L"-function of its complex conjugate character.