 Fourier representation of a smooth Selberg weight

If $f$ is smooth and supported on $[-1,1]$ and $g(t)={\int }_{\mathbb{R}}{e}^{x}f(x)e(-tx)dx$, Fourier inversion givesExpanding the square of the resulting Möbius-weighted divisor sum turns its mean value into an Euler product controlled by zeta functions near their pole at one.