  We study the numerical radius of Lipschitz operators on Banach spaces via the
Lipschitz numerical index, which is an analogue of the numerical index in
Banach space theory. We give a characterization of the numerical radius and
obtain a necessary and sufficient condition for Banach spaces to have Lipschitz
numerical index 1. As an application, we show that real lush spaces and
$C$-rich subspaces have Lipschitz numerical index 1. Moreover, using the
G$\hat{a}$teaux differentiability of Lipschitz operators, we characterize the
Lipschitz numerical index of separable Banach spaces with the RNP. Finally, we
prove that the Lipschitz numerical index has the stability properties for the
$c_0$-, $l_1$-, and $l_\infty$-sums of spaces and vector-valued function
spaces. From this, we show that the $C(K)$ spaces, $L_1(\mu)$-spaces and
$L_\infty(\nu)$ spaces have Lipschitz numerical index 1.
