  We give all possible holomorphic Eisenstein series on $\Gamma_0(p)$, of
rational weights greater than $2$, and with multiplier systems the same as
certain rational-weight eta-quotients at all cusps. We prove they are modular
forms and give their Fourier expansions. We establish four sorts of identities
that equate such series to rational-weight eta-quotients. As an application, we
give series expressions of special values of Euler Gamma function at any
rational arguments. These expressions involve exponential sums of Dedekind
sums.
