Is this a new characteristic function for the primes?

As $n\in\Bbb Z^+$, if $\chi_{\Bbb P}(n)=1$ then $$\chi_{\mathbb{P}}(n)=\frac{(-1)^{2\Gamma(n)/n}-1}{(-1)^{-2/n}-1}=\frac{(-1)^{2(n-1)!/n}-1}{(-1)^{-2/n}-1}=1\implies (-1)^{2(n-1)!/n}=(-1)^{-2/n}$$ which is equivalent to $$\frac{2(n-1)!}n\equiv-\frac2n\pmod2\implies2\cdot\left(\frac{(n-1)!+1}{n}\right)\equiv0\pmod2$$ which is true if the term in brackets is an integer; that is, if $n\mid (n-1)!+1$, which in turn is equivalent to Wilson's Theorem.

Note that on the other hand, $$\chi_{\Bbb P}(n)=0\implies\frac{2(n-1)!}n\equiv0\pmod2\implies n\mid(n-1)!$$ so $n$ cannot be prime.

Yes, this is a variant of Wilson's theorem.