How to see that $l^{1}(\mathbb{Z}) \to C(S^1, \mathbb{C}): a \mapsto \sum_{n \in \mathbb{Z}}a_n z^n$ is not surjective?

Question

Consider the map $$l^{1}(\mathbb{Z}) \to C(S^1, \mathbb{C}): a \mapsto (z \mapsto \sum_{n \in \mathbb{Z}}a_n z^n)$$

I want to show that this is not surjective.

I don't see which element I should take in the codomain that does not get attained. I feel like this somehow has a connection with basic Fourier theory. Thanks in advance!