Heuristic one-level density formula for logarithmic derivatives in cubic field families

Let $\frac 12 \leq \theta<\frac 56$ and $\omega\geq0$ be such that the stated local estimate holds. Let $\mathcal F^\pm(X)$ be the family of cubic fields of discriminant of sign $\pm$ and size at most $X$, with cardinality $N^\pm(X)$. For the associated Artin representations $f_K$, let $\theta_e$, $x_p$, $\beta_e(p)$, $C_1^\pm$, $C_2^\pm$, $A_3$, $A_4$, $\Gamma_\pm$, and $L$ be the quantities defined in the paper. There exists $0<\delta<\frac 16$ such that, for every fixed $\varepsilon>0$ and $r\in\mathbb C$ satisfying $\frac1L\ll\operatorname{Re}(r)<\delta$ and $|r|\leq X^{\varepsilon/2}$, an asymptotic formula for the family average of $L'(\frac12+r,f_K)/L(\frac12+r,f_K)$ is given by

$$\begin{aligned} \frac1{N^\pm(X)}\sum_{K\in\mathcal F^\pm(X)}\frac{L'(\frac12+r,f_K)}{L(\frac12+r,f_K)}={}&-\sum_{p,e\geq1}\left(\theta_e+\frac1p\right)\frac{x_p\log p}{p^{e(\frac12+r)}}\\ &-\frac{C_2^\pm}{C_1^\pm}X^{-\frac16}\left(1-\frac{C_2^\pm}{C_1^\pm}X^{-\frac16}\right)\sum_{p,e\geq1}\frac{(\beta_e(p)-p^{-e/3})\log p}{p^{e(\frac12+r)}}\\ &+\frac{C_2^\pm}{C_1^\pm}X^{-\frac16}\left(1-\frac{C_2^\pm}{C_1^\pm}X^{-\frac16}\right)\frac{\zeta'}\zeta\left(\frac56+r\right)\\ &-X^{-r}\frac{\Gamma_\pm(\frac12-r)}{\Gamma_\pm(\frac12+r)}\zeta(1-2r)\frac{A_3(-r,r)}{1-r}\\ &-\frac{C_2^\pm}{C_1^\pm}X^{-r-\frac16}\left(1-\frac{C_2^\pm}{C_1^\pm}X^{-\frac16}\right)\frac{\Gamma_\pm(\frac12-r)}{\Gamma_\pm(\frac12+r)}\zeta(1-2r)\\ &\qquad\times\left(\frac{\zeta(\frac56-r)}{\zeta(\frac56+r)}\frac{A_4(-r,r)}{1-\frac{6r}{5}}-\frac{A_3(-r,r)}{1-r}\right)+O_\varepsilon(X^{\theta-1+\varepsilon}). \end{aligned}$$

The one-level density conjecture. The displayed asymptotic formula should hold, and the two sums on its right-hand side are absolutely convergent.

The formula is a conjectural refinement of the family average of logarithmic derivatives, incorporating the lower-order terms of size $X^{-1/6}$ and the secondary zeta-factor contribution. Its status is not resolved in the supplied text.