  We explore the pairing state and gap structure of UTe$_2$ using a six-orbital
model which we call the $f$-$d$-$p$ model. Our model accurately reproduces the
quasi-two-dimensional Fermi surfaces consistent with recent de Haas-van Alphen
oscillation measurements and the $(0, \pm \pi, 0)$ antiferromagnetic spin
fluctuations observed by neutron scattering. We incorporate on-site Coulomb
repulsion for $f$ electrons and solve the linearized Eliashberg equation within
the third-order perturbation theory to investigate the superconducting symmetry
in UTe$_2$. The most likely state is found to be an $s$-wave state with a
highly anisotropic superconducting gap structure that exhibits a
point-node-like behavior of the specific heat at low temperatures.
