The Ratios Conjectures lemma for pair correlations of zeros

Let $f_d$ be a quadratic twist in the family and define

$$\mathcal{T}_{f_d}(\alpha,\beta,\gamma,\delta)=\int_0^T\frac{L(s+\alpha,f_d)L(1-s+\beta,\overline{f}_d)}{L(s+\gamma,f_d)L(1-s+\delta,\overline{f}_d)}\,dt,$$

where $s=1/2+it$. For $-1/4<\operatorname{Re}(\alpha),\operatorname{Re}(\beta)<1/4$, $1/\log(T)\ll\operatorname{Re}(\delta)<1/4$, and $\operatorname{Im}(\alpha),\operatorname{Im}(\beta)\ll_{\varepsilon}T^{1-\varepsilon}$ for all $\varepsilon>0$, define $Y_U$ and $A_L$ by the Euler products in the source. The Ratios Conjectures lemma.

$$\mathcal{T}_{f_d}(\alpha,\beta,\gamma,\delta)=\int_0^T\left(Y_U(\alpha,\beta,\gamma,\delta)A_L(\alpha,\beta,\gamma,\delta)+\left(\frac{\sqrt{M}|d|t}{2\pi}\right)^{-2(\alpha+\beta)}Y_U(-\beta,-\alpha,\gamma,\delta)A_L(-\beta,-\alpha,\gamma,\delta)\right)dt+O(T^{1/2+\varepsilon}).$$

The formula is used, under GRH, to evaluate pair-correlation statistics for zeros without a Fourier-transform support restriction. Its status is not resolved in the supplied text.