The Ratios Conjecture for quadratic twists of a modular form

Let $f$ be a holomorphic cusp form with associated quadratic-twist family $\mathcal{F}_f^{+}(X)$, and let $R_f(\alpha,\gamma)$ denote the average of the ratio of shifted $L$-functions over $d\in\mathcal{D}_f^{+}(X)$. For $-1/4 < \operatorname{Re}(\alpha) < 1/4$, $1/\log x \ll \operatorname{Re}(\gamma) < 1/4$, and $\operatorname{Im}(\alpha),\operatorname{Im}(\gamma) \ll X^{1-\varepsilon}$, the quantities $Y_f$, $\widetilde{Y}_f$, $A_f$, $\widetilde{A}_f$, and $\eta_f$ are defined by the Euler products and expectation specified in the conjecture statement. The Ratios Conjecture. The average satisfies

$$R_f(\alpha,\gamma)=\sum_{d\in\mathcal{D}_f^{+}(X)}\left[Y_fA_f(\alpha,\gamma)+\eta_f\left(\frac{\sqrt{M}|d|}{2\pi}\right)^{-2\alpha}\frac{\Gamma(k/2-\alpha)}{\Gamma(k/2+\alpha)}\widetilde{Y}_f\widetilde{A}_f(-\alpha,\gamma)\right]+O(X^{1/2+\varepsilon}),$$

with the local factors and the cases for $\eta_f$ exactly as displayed in the source. This conjectural formula is intended to provide the ratios input needed to derive the one-level density of zeros in the quadratic-twist family; its resolution is not supplied here.