What is the valid range of this formula?

Question

Let consider the Gaussian integral $$\int_{-\infty}^{\infty} dx \,e^{-\frac{1}{2}a\,x^2}=\sqrt\frac{2\pi}{a},\tag{1}$$ where $x$ is a real variable and $a$ is a positive real number. Now I would like to know, if I let $a$ to be complex, what is the valid range of this equality beyond which I need to do the analytical continuation? If possible, please show also the poles or branch cut singularities because of which equality (1) becomes invalid. Thank you very much.