  The purpose of this paper is to study $W(2,2)$ Lie conformal algebra, which
has a free $\mathbb{C}[\partial]$-basis $\{L, M\}$ such that $[L_\lambda
L]=(\partial+2\lambda)L$, $[L_\lambda M]=(\partial+2\lambda)M$, $[M_\lambda
M]=0$. In this paper, we study conformal derivations, central extensions and
conformal modules for this Lie conformal algebra. Also, we compute the
cohomology of this Lie conformal algebra with coefficients in its modules. In
particular, we determine its cohomology with trivial coefficients both for the
basic and reduced complexes.
