Ratios conjecture for cubic Dirichlet L-functions

Let $f$ be an even Schwartz function, let $w$ be the weight function, and define the weighted ratio average

$$R_w(\nu';\nu)=\sum_{\alpha}”' w\left(\frac{N(\alpha)}{X}\right)\frac{L\left(\chi_\alpha,\frac{1}{2}+\nu'\right)}{L\left(\chi_\alpha,\frac{1}{2}+\nu\right)}.$$

For $\operatorname{Re}(2\nu'+\nu)>-1$ and $\operatorname{Re}(\nu')>-1/2$, Ratios conjecture for cubic Dirichlet $L$-functions.

$$R_w(\nu';\nu)=W^*(X)\frac{\zeta\left(\frac{3}{2}+3\nu'\right)}{\zeta\left(\frac{3}{2}+2\nu'+\nu\right)}A(\nu';\nu)+O\left(X^{1/2+\epsilon}\right),$$

where

$$A(\nu';\nu)=\frac{\zeta\left(\frac{3}{2}+2\nu'+\nu\right)}{\zeta\left(\frac{3}{2}+3\nu'\right)}\prod_p\left(1+a(p)\frac{1-p^{\nu'-\nu}}{p^{3/2+3\nu'}-1}\right)$$

and equivalently

$$A(\nu';\nu)=\prod_{p\equiv1\mod 3}\left(1+O\left(p^{-5/2-2\nu'-\nu}+p^{-5/2-3\nu'}+p^{-3-4\nu'-2\nu}+p^{-2}\right)\right).$$

The ratios conjecture gives a precise prediction for averages of quotients of cubic Dirichlet $L$-functions and is used to predict their one-level density. The paper establishes this prediction in the setting considered, while the displayed assertion is presented as the conjectural input.