Solutions $\lambda$ of $\displaystyle 1-\lambda \mathcal{F}(e^{-ix})(\xi)=0$

Question

I am looking for reals $\lambda$ such that $$1-\lambda \mathcal{F}(e^{-ix})(\xi)=0,$$ where $\mathcal{F}$ is the Fourier transform. Using the Dirac distribution $\delta$, I found that the Fourier transform of $e^{-ix}$ is $\sqrt{2\pi}\delta(-1-\xi)$. So the equation becomes: $$1-\sqrt{2\pi}\lambda \delta(-1-\xi)=0.$$ Please help me to solve this equation. I thank you very much.