On $L$-functions of Hecke characters and anticyclotomic towers

In this paper, we generalize a work of Rohrlich. Let $K/\mathbb{Q}$ be an imaginary quadratic field and $\phi$ be a Hecke character of $K$ of infinite type (1,0) whose restriction to $\mathbb{Q}$ is the quadratic character corresponding to $K/\mathbb{Q}$. We consider a class of Hecke characters $\chi$, which are anticyclotomic twists of $\phi$ with ramification in a prescribed finite set of primes. We shall prove the central vanishing order of the Hecke $L$-function $L(s,\chi$) attached to each $\chi$ is 0 or 1 depending on the root number $W(\chi)$ for all but finitely many such $\chi$.