  In this paper we establish a coarse Jacquet-Zagier trace identity for
GL$(n).$ We prove the absolute convergence when $\Re(s)>1$ and $0<\Re(s)<1;$
and obtain holomorphic continuation under almost all character twist. Moreover,
as an application, we prove that holomorphy of certain adjoint $L$-functions
for GL$(n)$ implies Dedekind conjecture of degree $n$. Some nonvanishing
results are also discussed.
