Let $X \sim \mathcal{N}(0, \sigma^2)$. Define the sub-exponential norm as follows for a random variable $Z$: $$ \|Z\|_{\psi_1} := \inf \left\{k>0 \vert \mathbb{E}\left[\exp \frac{|Z|}{k}\right] \leq 2\right\}. $$
I would like to explicitly compute this norm for $X$. I have done this for the sub-gaussian norm by simply computing the expectation, which is monotonically increasing, and setting it equal to $2$, but this approach doesn't really work here because I can't seem to get a useable result for the integral. I think the norm should still be of the form $C\sigma$ for a positive constant $C$ (or at least something similar), but I can't seem to figure out how it looks exactly. Any help would be appreciated.