Is the half iterate of $2\sinh(x)$ - expanded from fixed point $0$ - analytic near the real line $\mathbb{R}$?

Question

Is the half iterate of $2\sinh(x)$ - expanded from fixed point $0$ - analytic near the real line $\mathbb{R}$?

I know this function has 2 other fixed points apart from $0$, so I'm not sure.

Also does analytic at some place imply it satisfies its functional equation(being half-iterate of $2\sinh(x)$) at that place ?