General critical-value rationality conjecture for degree-six L-functions

Let $F$ be the totally real field with set of archimedean places $\Sigma_{\infty}$, and let $\pi$ and $\tau$ be the automorphic representations considered in the paper, with balanced and unbalanced archimedean places $\Sigma_b$ and $\Sigma_{ub}$, respectively. Write $\kappa_0$, $\kappa$, $D_F$, $u(\varepsilon,\pi)$, and $\nu^{\Sigma_{ub}}(\pi)$ and $\nu^{\Sigma_b}(\tau)$ for the weights, discriminant, Shimura invariant, and Harris periods introduced in the paper. For a critical half-integer $m+\frac{1}{2}$, let $\varepsilon=(\varepsilon_v)$ satisfy $\varepsilon_v\equiv m+\kappa_0+1\pmod 2$ for every $v$. The general critical-value rationality conjecture. For every $\sigma\in\operatorname{Aut}(\mathbb{C})$, one has

$$\left(\frac{L\left(m+\frac{1}{2},\operatorname{Sym}^{2}\tau\times\pi\right)}{\pi^{|\kappa_0-\kappa+1|+|\Sigma_b|}\sqrt{D_F}|u(\varepsilon,\pi)|\nu^{\Sigma_{ub}}(\pi)\nu^{\Sigma_b}(\tau)^2}\right)^\sigma= \frac{L\left(m+\frac{1}{2},\operatorname{Sym}^{2}\tau^\sigma\times\pi^\sigma\right)}{\pi^{|\kappa_0-\kappa+1|+|\Sigma_b|}\sqrt{D_F}|u(\varepsilon,\pi^\sigma)|\nu^{\Sigma_{ub}}(\pi^\sigma)\nu^{\Sigma_b}(\tau^\sigma)^2}.$$

This conjecture extends the rationality results proved in the purely balanced and purely unbalanced cases to arbitrary mixtures of balanced and unbalanced places, and is motivated by Deligne's conjecture on special values and explicit calculations of the relevant periods. Its status is not resolved in the supplied source.