Inversion formula of Dirichlet polynomials and the approximate functional equation of Dirichlet's L-functions

In this note, we prove an inversion formula for Dirichlet polynomials involving a non-principal character. By using Heath-Brown's upper-bound for Dirichlet's L-functions, we are able to obtain a non-trivial upper-bound for the error term in the q aspect. As an application we obtain an approximate functional equation for Dirichlet's L-functions to the right of the critical line with a sharp error term.