The recipe conjecture for averages of quadratic twists of Dirichlet polynomials

Let $cal D$ be a family of fundamental discriminants in a fixed arithmetic progression, let $cal M_A(D;\ell)$ denote the corresponding smoothed average of $\mathcal L_A(1/2,\chi_d)\chi_d(\ell)$, and let $X_d$ and $\mathcal B^{(d)}$ be defined by the functional equation and Dirichlet series in the setup. For a set $A$ of complex numbers with positive real parts $\ll 1/\log D$ and imaginary parts $\ll D$, write $A-U+U^- = \{a\in A:a\notin U\}\cup\{-u:u\in U\}$. The recipe conjecture. There is a $\delta>0$, depending on the number of elements of $A$, such that

$$\mathcal M_A(D;\ell)= \sum_{d\in \mathcal D\atop (d,\ell)=1} \Psi\left(\frac dD\right) \sum_{U\subset A} \prod_{u\in U} X_d(1/2 +u) \mathcal B^{(d)}(A-U+U^-;\ell)+O(\ell^{1/2}D^{1-\delta}).$$

This is the CFKRS recipe for the family of quadratic characters in a fixed arithmetic progression. It predicts the main terms obtained by applying the functional equations independently to subsets of the shifts, with the coprimality condition built into $\mathcal B^{(d)}$; the paper treats it as conjectural.