We present an incomplete LU preconditioner for solving discretized Helmholtz problems. The preconditioner is based on an analytic factorization of the Helmholtz operator. This allows us to take the physical properties of the acoustics problem modeled by the Helmholtz equation into account in the preconditioner. We show how the parameters in the preconditioner can be chosen in order to make it effective. Numerical experiments show that the new preconditioner leads to convergent iterative methods even for large wave numbers, and it outperforms classical ILU preconditioners by a large margin.