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In this paper, we study the following stochastic heat equation \[  \partial_tu=\mathcal{L} u(t,x)+\dot{B},\quad  u(0,x)=0,\quad 0\le t\le T,\quad x\in\mathbb{R}d, \] where $\mathcal{L}$ isthe generator of a L\'evy process $X$ taking value in $\mathbb{R}^d$, $B$ is afractional-colored Gaussian noise with Hurst index$H\in\left(\frac12,\,1\right)$ for the time variable and spatial covariancefunction $f$ which is the Fourier transform of a tempered measure $\mu.$  After establishing the existence of solution for the stochastic heatequation, we study the regularity of the solution $\{u(t,x),\, t\ge 0,\,x\in\mathbb{R}^d\}$ in both time and space variables. Under mild conditions, wegive the exact uniform modulus of continuity and a Chung-type law of iteratedlogarithm for the sample function $(t,x)\mapsto u(t,x)$. Our results generalizeand strengthen the corresponding results of Balan and Tudor (2008) and Tudorand Xiao (2017).

Sharp Space-Time Regularity of the Solution to Stochastic Heat Equation Driven by Fractional-Colored Noise