In this paper we apply the $L$-function Ratios Conjecture to compute the one-level density for a symplectic family of $L$-functions attached to Hecke characters of infinite order. When the support of the Fourier transform of the corresponding test function $f$ reaches $1$, we observe a transition in the main term, as well as in the lower order term. Assuming GRH, we then directly calculate main and lower order terms for test functions $f$ such that supp($\widehat{f}) \subset (-1,1)$, and observe that the result is in agreement with the prediction provided by the Ratios Conjecture. As a corollary, we deduce that, under GRH, at least $75\%$ of these $L$-functions do not vanish at the central point.

中文翻译：

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Hecke L 函数辛族的单级密度的低阶项

在本文中，我们应用$L$-函数比率猜想来计算附加到无限阶Hecke 字符的$L$-函数辛族的一级密度。当对应测试函数 $f$ 的傅立叶变换的支持度达到 $1$ 时，我们观察到主项和低阶项的转换。假设GRH，我们然后直接计算测试函数$f$的主项和低阶项，使得supp($\widehat{f})\subset(-1,1)$，并观察结果与预测一致由比率猜想提供。作为推论，我们推断，在 GRH 下，这些 $L$ 函数中至少有 $75\%$ 不会在中心点消失。