Arithmeticity of critical $L$-values for Hilbert-Siegel modular forms in the left half plane and $s = 1$

For a cuspidal automorphic representation $π_f$ generated by a Hilbert-Siegel modular form $f$ and a Hecke character $μ$ with parallel signature, we show the arithmeticity of critical $L$-values of the standard $L$-function $L^S(s,π_f, μ)$ twisted by $μ$ in the left half plane. As a consequence, we