  In this paper we propose new Z-type nonlinear weights of the fifth-order
weighted essentially non-oscillatory (WENO) finite difference scheme for
hyperbolic conservation laws. Instead of employing the classical smoothness
indicators for the nonlinear weights, we take the $p$th root of the smoothness
indicators and follow the form of Z-type nonlinear weights, leading to fifth
order accuracy in smooth regions, even at the critical points, and sharper
approximations around the discontinuities. We also prove that the proposed
nonlinear weights converge to the linear weights as $p \to \infty$, implying
the convergence of the resulting WENO numerical flux to the finite difference
numerical flux. Finally, numerical examples are presented by comparing with
other WENO schemes, such as WENO-JS, WENO-M and WENO-Z, to demonstrate that the
proposed WENO scheme performs better in shock capturing.
