Ratios-model formula for the averaged one-level density of Hecke L-functions

Let $\mathcal{F}(K)=\{L_k(s):1\le k\le K\}$, let $M=\log K$, and let $R_K(\alpha,\gamma)$ be the ratios-conjecture prediction built from the Euler product $G(\alpha,\gamma)$. For shifts satisfying the constraints in the ratios domain, the ratios-model one-level-density conjecture.

$$R_K(\alpha,\gamma)=G(\alpha,\gamma)+\frac{1}{1-2\alpha}\left(\frac{\pi}{2K}\right)^{2\alpha}G(-\alpha,\gamma)+O(K^{-\frac12+\epsilon}).$$

Here

$$G(\alpha,\gamma)=\left(1-\frac{1}{2^{1+\gamma+\alpha}}\right)\prod_{p\equiv1(4)}\left(1-\frac{2}{p^{1+\gamma+\alpha}}+\frac{1}{p^{1+2\gamma}}\right)\prod_{p\equiv3(4)}\left(1-\frac{1}{p^{1+2\gamma}}\right)\prod_p\left(1-\frac{1}{p^{1+2\alpha}}\right)^{-1}.$$

This is an intermediate ratios-conjecture prediction used to obtain lower-order terms in the one-level density; the source gives no independent resolution status.