I tried to prove the standard identities of the Dedekind eta function $$\eta(\tau)=q^{\frac{1}{24}} \prod_{n=1}^\infty (1-q^n),$$ where $q=\exp(2\pi i \tau)$ for some complex number $\tau$, but encountered some challenges.
First, $$\eta(-\frac{1}{\tau}) =\sqrt{-i\tau}\eta (\tau) $$
I think there's something to do with the $\Gamma$ functions. However, I'm not really sure how to convert the $-\frac{1}{\tau}$ on the exponential back to the $\tau$, and extract a $\sqrt{i\tau}$ on the front.
Second, $$|\tau|^{1/2} |\eta(\tau)|^2$$ is invariant under $\tau\rightarrow -\frac{1}{\tau}$, where $|\eta(\tau)|^2 = \eta(\tau) \cdot \eta(\bar \tau)$. I took it from the papers for granted, but when I tried to prove it, I found it was somehow confusing given the first identities.
Could you show how to prove the above two identities, please? I was working on the line of the series expansion through the Euler identities, but it didn't work.