L-函数在中心点的值是否为零具有深刻的算术意义， 是当前数论领域的研究前沿之一。从解析数论的角度出发，在此方面的研究目前有两种主要途径，一是推导L-函数簇中心点均值及高次幂的渐进表示式， 另一是推导L-函数簇的一阶低零点密度。尽管通过上述两种途径已有大量工作围绕经典的狄利克雷L-函数以及自守L-函数开展，但是关于数域上的Hecke L-函数的研究还方兴未艾。本项目在申请人已有的研究基础上，研究数域上若干Hecke L-函数在中心点均值的渐进表示式以及若干Hecke L-函数簇的一阶低零点密度表示式。 作为应用，我们将给出在中心点值非零的Hecke L-函数的数量的估计。在此方面的研究结果将加深我们对数域上的Hecke L-函数在中心点取值的了解。

The non-vanishing problem of central values of L-funtions has important arithmetic meanings, and hence is one of the current research frontiers. From the point view of analytic number theory, there are currently two main approaches towards the non-vanishing problem. One is to study moments of families of L-functions while the other is to study the one level density of low-lying zeros of families of L-functions. Although there has been a lot of work done on the classical Dirichlet L-functions as well as automorphic L-functions in the literature via the above mentioned two approaches, little has been done on Hecke L-functions of number fields. In this project, we plan to study moments and one level densities of various Hecke L-functions so that we can use the results to estimate the number of L-functions with no-nvanishing central values. The results will enhance our understadning on the zeros of the Hecke L-functions.