Upper bounds on L(1,χ) taking into account ramified prime ideals

Let χ range over the non-trivial primitive characters associated with the abelian extensions L/K of a given number field K, i.e. over the non-trivial primitive characters on ray class groups of K. Let fχ be the norm of the finite part of the conductor of such a character. It is known that |L(1,χ)|≤12Ress=1(ζK(s))log⁡fχ+O(1), where the implied constants in this O(1) are effective and depend on K only. The proof of this result suggests that one can expect better upper bounds by taking into account prime ideals of K dividing the conductor of χ, i.e. ramified prime ideals. This has already been done only in the case that K=Q. This paper is devoted to giving for the first time such improvements for any K. As a non-trivial example, we give fully explicit bounds when K is an imaginary quadratic number field.