Consider $E_1:\mathbb{C}\rightarrow \mathbb{C}, s\mapsto(1-s)e^s$ and the infinite (and on $\mathbb{C}$ holomorphic) product: $$\Pi_k(s)=\prod_{q \in \mathbb{Z}^\times\setminus{k}} E_1\left(\frac sq\right)$$ Im trying to find the Fourier transform of this product. For sake of argument let us use the transform defined by: $$\mathcal{F}[f](x)=\int_{-\infty}^\infty f(t)e^{-i \,2 \pi x \,t} dt$$ My first attempt was to use the convolution theorem that establishes a duality: $$\mathcal{F}[f\cdot g]=\mathcal{F}[f]*\mathcal{F}[g]$$ Where $*$ denotes the convolution of functions.
We now have: \begin{align*} \mathcal{F}\left[\Pi_k\right]&=\mathcal{F}\left[\prod_{q \in \mathbb{Z}^\times\setminus\{k\}} E_1\left(\frac sq\right)\right]\\ &=\prod_{q\in\mathbb{Z}^\times\setminus\{k\}}^{\text{conv.}} \mathcal{F}\left[E_1\left(\frac sq\right)\right] \end{align*} This is where I get stuck since $E_1$ doesn't seem to have a well definied Fourier transform due to the factor $e^s$ of $E_1$. Any ideas how I should proceed? Thanks in advance!