CFKRS–DGH twisted moment conjecture for quadratic Dirichlet L-functions

Let $X$ be large, let $\Phi$ be the smooth test function used in the moment, and let $\alpha$ be complex. For an odd squarefree integer $l$, define

$$M(\alpha,l)=\sideset{}{^*}\sum_{(d,2)=1}\chi_{8d}(l)L\left(\tfrac12+\alpha,\chi_{8d}\right)\Phi\left(\frac dX\right).$$

Let $\widetilde{\Phi}$ be the Mellin transform of $\Phi$, let $\zeta_2(s)=(1-2^{-s})\zeta(s)$, and define $B_\alpha(l)$ by

$$\zeta_2(1+2\alpha)B_\alpha(l)=\sum_{(n,2)=1}\frac{1}{n^{1+2\alpha}}\prod_{p\mid nl}(1+p^{-1})^{-1}.$$

CFKRS–DGH twisted moment conjecture. Uniformly for $|\operatorname{Re}(\alpha)|\ll(\log X)^{-1}$ and $|\operatorname{Im}(\alpha)|\ll(\log X)^2$, one has

$$M(\alpha,l)=\frac{X\widetilde{\Phi}(1)}{2\zeta_2(2)}l^{-1/2-\alpha}\zeta_2(1+2\alpha)B_\alpha(l)+\frac{X^{1-\alpha}\widetilde{\Phi}(1-\alpha)\gamma_\alpha}{2\zeta_2(2)}l^{-1/2+\alpha}\zeta_2(1-2\alpha)B_{-\alpha}(l)+O\left((lX)^{1/2+\varepsilon}\right).$$

This twisted asymptotic is the form needed for mollification and amplification; the source presents it after the untwisted conjecture and attributes the related moment recipe to CFKRS and the twisted formulation to HY. Its resolution is not specified in the paper.