Linear recurrences and rational Lambert series

For a sequence $\gamma =({\gamma }_n{)}_{n\ge 1}$, define $$L_{\gamma }(z):=\sum _{n\ge 1}{\gamma }_n\frac{z^n}{1-z^n}=\sum _{n\ge 1}\left(\sum _{d\mid n}{\gamma }_d\right)z^n\mathrm{.}$$ We prove a short rigidity theorem: if $\gamma$ is eventually linearly recurrent and $L_{\gamma }(z)$ is rational, then $\gamma$ is finitely supported. Equivalently, among sequences with rational ordinary generating function, the only ones whose Lambert series is rational are the finitely supported sequences. The proof specializes the data at a finite place of a finitely generated ring and then uses the periodicity of recurrences over finite fields.