By using periodic functions from the nonnegative integers to the complex numbers, we generalize the generating function of the $q$-Apostol type Eulerian polynomials and numbers attached the character defined in [1]. Then using this generating function, we a construct new $L$-type series. By using periodic functions, we derive decomposition of the generating functions for the $q$-Euler numbers and polynomials. Applying the Mellin transformation to the decomposition of the generating functions, we introduce and investigate the various properties of a certain new family of the Dirichlet type $L$-series and the Dirichlet $L$-function. Finally, we derive many potentially useful results involving these functions polynomials and numbers.