On L-functions of Hecke characters and anticyclotomic towers

We generalize a result of Rohrlich (1984). Let K/Q be an imaginary quadratic field and ϕ be a Hecke character of K of infinite type (1,0) whose restriction to Q is the quadratic character corresponding to K/Q. We consider a class of Hecke characters χ, which are anticyclotomic twists of ϕ with ramification in a prescribed finite set of primes. We prove that the central vanishing order of the Hecke L-function L(s,χ) attached to each χ is 0 or 1 depending on the root number W(χ) for all but finitely many such χ.