Adelic conjecture for local zeta functions and string amplitudes

Fix $N$ and let $Z_{\mathbb{R}}^{(N)}(\boldsymbol{s})$ and $Z_{\mathbb{Q}_p}^{(N)}(\boldsymbol{s})$ denote the archimedean and $p$-adic local zeta functions, respectively, regarded through their meromorphic continuations in $\boldsymbol{s}\in\mathbb{C}^{D}$. Adelic local-zeta conjecture. There is a meromorphic function $\Gamma(\boldsymbol{s})$ on $\mathbb{C}^{D}$ such that

$$Z_{\mathbb{R}}^{(N)}(\boldsymbol{s})=\Gamma(\boldsymbol{s})\prod_{p<\infty}Z_{\mathbb{Q}_p}^{(N)}(\boldsymbol{s})$$

for every $\boldsymbol{s}\in\mathbb{C}^{D}$. This proposes an adelic relation between the real and non-archimedean local zeta functions, extending the type of product relations familiar from adelic string amplitudes; its resolution is not given in the source.