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Heavy fermion criticality has been a long-standing problem in condensed matter physics. Here we study a one-dimensional Kondo lattice model through numerical simulation and observe signatures of local criticality. We vary the Kondo couplingJ_K$J_K$at fixed doping x. At large positiveJ_K$J_K$, we confirm the expected conventional Luttinger liquid phase with2k_F=\frac{1+x}{2}$2k_F=\frac{1+x}{2}$(in units of2\pi$2\pi$), an analogue of the heavy Fermi liquid (HFL) in the higher dimension. In theJ_K ≤ 0$J_K\le 0$side, our simulation finds the existence of a fractional Luttinger liquid (LL\star$\star$) phase with2k_F=\frac{x}{2}$2k_F=\frac{x}{2}$, accompanied by a gapless spin mode originating from localized spin moments, which serves as an analogue of the fractional Fermi liquid (FL\star$\star$) phase in higher dimensions. The LL\star$\star$phase becomes unstable and transitions to a spin-gapped Luther-Emery (LE) liquid phase at small positiveJ_K$J_K$. Then we mainly focus on the “critical regime” between the LE phase and the LL phase. Approaching the critical point from the spin-gapped LE phase, we often find that the spin gap vanishes continuously, while the spin-spin correlation length in real space stays finite and small. For a certain range of doping, in a point (or narrow region) ofJ_K$J_K$, the dynamical spin structure factor obtained through the time-evolving block decimation (TEBD) simulation shows dispersion-less spin fluctuations in a finite range of momentum space above a small energy scale (around0.035 J$0.035J$) that is limited by the TEBD accuracy. All of these results are unexpected for a regular gapless phase (or critical point) described by conformal field theory (CFT). Instead, they are more consistent with exotic ultra-local criticality with an infinite dynamical exponentz=+$z=+$. The numerical discovery here may have important implications on our general theoretical understanding of the strange metals in heavy fermion systems. Lastly, we propose to simulate the model in a bilayer optical lattice with a potential difference.