2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/157278The validity of the CFT in the Tomonaga-Luttinger liquid with the $1/r^β$ type long-range interactions is discussed. The arguments by CFT for long-range forward scatterings predict the finite size corrections which depend on the power of the long-range interactions. Especially from finite size corrections in the ground state, we find the effective central charge deviates from $c=1$. These conformal arguments are numerically tested in the one particle excitations and the ground state.This paper is the final version. 23 pages, 11 figuresStrongly Correlated Electronstext