On the length equalities for one-dimensional rings

In this article we characterize noetherian local one-dimensional analytically irreducible and residually rational domains (R,m_R) which are non-Gorenstein, the non-negative integer is equal to τ_R-1 and ?(R/(C+xR))=2, where τ_R is the Cohen-Macaulay type of R, C is the conductor of R in the integral closure of R in its quotient field Q(R) and xR is a minimal reduction of m by giving some conditions on the numerical semi-group v(R) of R.