Does absolute integrability imply square integrability or are these independent properties (for functions in Complex Fields)?

Given a function f(x) : $\mathbb{R} \to \mathbb{C}$

f(x) is said to be absolute integrable iff $\int_{-\infty}^\infty |f(x)|dx$ is finite.f(x) is said to be square integrable iff $\int_{-\infty}^\infty |f(x)|^2dx$ is finite.

I know square integrability does not imply absolute integrability as f(x) = $x^{-\frac{1}{2}}u(x-1)$ (where u(x) is the unit step function) is square integrable but not absolute integrable.I cant find an example of a function which is absolute integrable but not square integrable. Intuitively it makes sense why such a function is not possible but I want a proof. Also does the function does having finite maximas, minimas and discontinuities change anything?