Special value formulae of Rankin -Selberg L-functions

In this paper, we prove a special value formula of level N of Rankin-Selberg L-function associated to a Hilbert modular form of higher weight and a ring class character of an totally imaginary quadratic extension of a totally real field. The formula relates the special value of the Rankin-Selberg L-functions at s = ½ to the value of certain test form at some CM-point on a 0-dimensional Shimura Variety associated to a quaterion algebra. The formula generalizes the formula proved by Shou-Wu Zhang which is a vast generalization of classical Gross-Zagier formula. The proof is based on a formula (level ND) of Hui Xue combined with a technique of Eisenstein Series to compute the universal constants which arise in the comparison of both formulae of level N and ND