Suppose $X_1, \ldots, X_n$ are iid $\mathrm{N}(0, \exp (2 \gamma))$; that is, the density of $X_i$ is $$ (2 \pi)^{-1 / 2} e^{-\gamma} \exp \left(-x^2 e^{-2 \gamma} / 2\right) . $$
I want to calculate the fisher information of $\gamma$,but the result I get is 2$n$,which means the fisher information has nothing to do with $\gamma$. I find it confusing, Could anyone help me? Please