Fast multi-precision computation of some Euler products

For every modulus $q\ge3$, we define a family of subsets $\mathcal{A}$ of the multiplicative group $(\mathbb{Z}/{q}\mathbb{Z})^\times$ for which the Euler product $\prod_{p\text{mod}q\in\mathcal{A}}(1-p^{-s})$ can be computed in double exponential time, where $s>1$ is some given real number. We provide a Sage script to do so, and extend our result to compute Euler products $\prod_{p\in\mathcal{A}}F(1/p)/G(1/p)$ where $F$ and $G$ are polynomials with real coefficients, when this product converges absolutely. This enables us to give precise values of several Euler products intervening in Number Theory.