We report a $^{13}$C-NMR study on the ambient-pressure metallic phase of the layered organic conductor $theta$-(BEDT-TTF)$_{2}$I$_{3}$ [BEDT-TTF: bisethylenedithio-tetrathiafulvalene], which is expected to connect the physics of correlated electrons and Dirac electrons under pressure. The orientation dependence of the NMR spectra shows that all BEDT-TTF molecules in the unit cell are to be seen equivalent from a microscopic point of view. This feature is consistent with the orthorhombic symmetry of the BEDT-TTF sublattice and also indicates that the monoclinic $I_{3}$ sublattice, which should make three molecules in the unit cell nonequivalent, is not practically influential on the electronic state in the conducting BEDT-TTF layers at ambient pressure. There is no signature of charge disproportionation in opposition to most of the $theta$-type BEDT-TTF salts. The analyses of NMR Knight shift, $K$, and the nuclear spin-lattice relaxation rate, $1/T_{1}$, revealed that the degree of electron correlation, evaluated by the Korringa ratio [$varpropto 1/(T_{1}TK^{2}$)], is in an intermediate regime. However, NMR relaxation rate $1/T_{1}$ is enhanced above $sim$ 200K, which possibly indicates that the system enters into a quantum critical regime of charge-order fluctuations as suggested theoretically.
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