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

For a cuspidal automorphic representation ${\pi }_f$ generated by a Hilbert-Siegel modular form $f$ and a Hecke character $\mu$ with parallel signature, we show the arithmeticity of critical $L$-values of the standard $L$-function $L^S(s,{\pi }_f,\mu )$ twisted by $\mu$ in the left half plane. As a consequence, we show the arithmeticity at $s=1$ under the assumption that there exists a nonarchimedean place $v$ such that ${\pi }_v$ is unramified and ${\mu }_v$ is ramified.