The ratios conjecture for the second family of elliptic curves

Let ${\mathcal F}(X)$ be the second family of elliptic curves under consideration, with conductor parameter $C(t)$. Define

$$Y(\alpha,\gamma):=\frac{\zeta(1+2\gamma)\zeta(1+\gamma)}{\zeta(1+\alpha+\gamma)\zeta(1+\alpha)}$$

and let $A(\alpha,\gamma)$ be the analytic Euler product specified by the factorization of $H(\alpha,\gamma)$. Let $\varepsilon>0$, with ${\rm Re}(\alpha)>-1/4$, ${\rm Re}(\gamma)\gg 1/\log X$, and ${\rm Im}(\alpha),{\rm Im}(\gamma)\ll_\varepsilon X^{1-\varepsilon}$. Ratios Conjecture.

$$\begin{aligned} \frac{1}{|{\mathcal F}(X)|}\sum_{E_t\in{\mathcal F}(X)}\frac{L(\frac12+\alpha,E_t)}{L(\frac12+\gamma,E_t)} ={}&\frac{1}{|{\mathcal F}(X)|}\sum_{E_t\in{\mathcal F}(X)}\left[Y(\alpha,\gamma)A(\alpha,\gamma) -\left(\frac{\sqrt{C(t)}}{2\pi}\right)^{-2\alpha}\frac{\Gamma(1-\alpha)}{\Gamma(1+\alpha)}Y(-\alpha,\gamma)A(-\alpha,\gamma)\right]\\ &+O(X^{-1/2+\varepsilon}). \end{aligned}$$

This is the ratios prediction for the second elliptic-curve family, whose arithmetic factor contains the additional zeta factors arising from the character $\chi_4$. The stated error term and shift range are conjectural.