Conrey–Farmer–Keating–Rubinstein–Snaith shifted-moment conjecture for elliptic-curve twists

Let $k,\beta\in\mathbb{N}$ and let $\underline{\theta}=(\theta_1,\dots,\theta_k)\in\mathbb{R}^k$. For the even-sign quadratic twists of an elliptic curve $E/\mathbb{Q}$, let $Y_d(s)=|d|^{1-2s}Y(s)$ be the functional-equation factor and let $\mathcal{L}_p$ denote the local Euler factors of $L_E$. Conrey–Farmer–Keating–Rubinstein–Snaith's shifted-moment conjecture. There exists $\delta>0$ such that

$$\frac{1}{D^*}\sideset{}{^*}\sum_{\substack{|d|\leq D\w_E\chi_d(-M)=1}}\prod_{m=1}^kL_E(\tfrac12+i\theta_m,\chi_d)^{2\beta}=\frac{1}{D^*}\sideset{}{^*}\sum_{\substack{|d|\leq D\w_E\chi_d(-M)=1}}\prod_{m=1}^kY_d(\tfrac12+i\theta_m)^\beta\Upsilon_{k,\beta}(\log|d|,\underline{\theta})+O(D^{-\delta}),$$

where $\Upsilon_{k,\beta}$, $H$, and the Euler product $B_{k\beta}$ are exactly as specified in the source statement. The conjecture is invoked to derive the Bailey–Keating elliptic-curve-twist conjecture and is not proved in the source.