How to get some insight in the following integral:

\begin{equation} \mathcal{I}(s)=\int_0^\infty x^{-x}e^{sx}\text{d} x \end{equation}

where $s$ is real (and the lower integration bound may be set to $a>0$)?

Alternatively, can we estimate this kind of series:

\begin{equation} \mathcal{S}(s)=\sum_{n=1}^\infty n^{-n}e^{sn} \end{equation} where $s$ is real?