The electronic and superconducting properties associated with the topologically non-trivial bands in Weyl semimetals have recently attracted much attention. We report the microscopic properties of the type-I Weyl semimetal TaAs measured by $^{75}$As nuclear magnetic (quadrupole) resonance under zero and elevated magnetic fields over a wide temperature range up to 500 K. The magnetic susceptibility measured by the Knight shift $K$ is found to be negative at low magnetic fields and have a strong field ($B$) dependence as ln$B$ at $T$ = 1.56 K. Such nonlinear field-dependent magnetization can be well accounted for by Landau diamagnetism arising from the 3D linearly dispersed bands, and thus is a fingerprint of topological semimetals. We further study the low-energy excitations by the spin-lattice relaxation rate 1/$T_{1}$. At zero field and 30 K $leq Tleq$ 250 K, 1/$T_{1}T$ shows a $T^{2}$ variation due to Weyl nodes excitations. At $B sim$ 13 T, $1/T_1T$ exhibits the same $T$-dependence but with a smaller value, scaling with $K^2propto T^2$, which indicates that the Korringa relation also holds for a Weyl semimetal. Analysis of the Korringa ratio reveals that the energy range of the linear bands is about 250 K in TaAs.
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