Which role does the $\frac{1}{24}$ in the Dedekind $\eta$-function play?

Question

The Dedekind $\eta$-function is defined as $$\eta(z) = q^{\frac{1}{24}} \prod_{n = 1}^\infty (1 - q^n)^{-1}$$ where $q = e^{2 \pi i z}$.

My question is: If I start with the Euler-product $\prod_{n = 1}^\infty (1 - q^n)^{-1}$, how do I come to the point where multiplication with $q^{\frac{1}{24}}$ makes sense?

Thanks!

Answer 1

One reason is, that the modular discriminant $Δ(z) = \eta (z)^{24}$ is a modular form of weight $12$. The $\mathbb{C}$-vector space of cusps forms of weight $12$ and level $1$ has dimension $1$, i.e., $\dim S_{12}(SL_2(\mathbb{Z}))=1$, and $\eta (z)^{24}$ is a generator. The presence of $24$ here can also be connected to the Leech lattice, which has $24$ dimensions.