In this paper, the method of inverse spectral problem is used to integrate the nonlinear Hirota equation in the class of periodic infinite-zone functions. An evolution of the spectral data of the periodic Dirac operator is introduced, where the coefficient of this operator is the solution of the nonlinear Hirota equation. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five times continuously differentiable periodic infinite-zone functions is shown. In addition, it is proved that if the initial function is a
π
\pi
-periodic real-analytic function, then the solution of the Cauchy problem for the Hirota equation is also a real-analytic function in the variable
x
x
; and if the number
π
/
2
\pi /2
is a period (antiperiod) of the initial function, then the number
π
/
2
\pi /2
is a period (antiperiod) in the variable
x
x
of the solution of the Cauchy problems for the Hirota equation.