  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 $\mathcal{L}$ is
the generator of a L\'evy process $X$ taking value in $\mathbb{R}^d$, $B$ is a
fractional-colored Gaussian noise with Hurst index
$H\in\left(\frac12,\,1\right)$ for the time variable and spatial covariance
function $f$ which is the Fourier transform of a tempered measure $\mu.$
  After establishing the existence of solution for the stochastic heat
equation, 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, we
give the exact uniform modulus of continuity and a Chung-type law of iterated
logarithm for the sample function $(t,x)\mapsto u(t,x)$. Our results generalize
and strengthen the corresponding results of Balan and Tudor (2008) and Tudor
and Xiao (2017).
