Irrationality of $\min$ and $\arg\,\min$ of $\Gamma|_{[1, 2]}$

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The Gamma function achieves a local minimum at $x^* \approx 1.46163$, where $\Gamma(x^*) \approx 0.88560$.

Can $x^*$ and $\Gamma(x^*)$ easily be proven irrational? Are they transcendental?