Conjecture for the average logarithmic derivative of quadratic Dirichlet L-functions

Let $\chi_n$ denote the quadratic Dirichlet character associated with an odd squarefree integer $n$, let $\mu$ be the Möbius function, and let $P_{D,2}(\alpha,\beta)$, $\Gamma_e$, and $\Gamma_o$ be the factors used in the ratios conjecture. Suppose that

$$\frac1{\log X}\ll\operatorname{Re}(r)\ll\frac14,\qquad \operatorname{Im}(r)\ll X^{1-\varepsilon}.$$

The logarithmic-derivative conjecture. Then

$$\begin{aligned} \sum_{n\leq X}\frac{\mu^2(n)L'(1/2+r,\chi_n)}{L(1/2+r,\chi_n)}={}&\frac{2X}{3\zeta(2)}\left(\frac{\zeta'(1+2r)}{\zeta(1+2r)}+\sum_{p>2}\frac{\log p}{(p+1)(p^{1+2r}-1)}\right)\\ &-\frac{X^{1-r}\pi^r\left(\Gamma_e\left(\frac12+r\right)+\Gamma_o\left(\frac12+r\right)\right)\zeta(1-2r)}{(1-r)3\zeta(2)}P_{D,2}(-r,r)\\ &+O(X^{1/2+\varepsilon}). \end{aligned}$$

The conjecture is obtained by differentiating the preceding ratios conjecture at equal shifts and predicts an asymptotic for averaged logarithmic derivatives of quadratic Dirichlet $L$-functions.