  In this paper, we study the dual of generalized bent functions $f:
V_{n}\rightarrow \mathbb{Z}_{p^k}$ where $V_{n}$ is an $n$-dimensional vector
space over $\mathbb{F}_{p}$ and $p$ is an odd prime, $k$ is a positive integer.
It is known that weakly regular generalized bent functions always appear in
pairs since the dual of a weakly regular generalized bent function is also a
weakly regular generalized bent function. The dual of non-weakly regular
generalized bent functions can be generalized bent or not generalized bent. By
generalizing the construction of \cite{Cesmelioglu5}, we obtain an explicit
construction of generalized bent functions for which the dual can be
generalized bent or not generalized bent. We show that the generalized indirect
sum construction method given in \cite{Wang} can provide a secondary
construction of generalized bent functions for which the dual can be
generalized bent or not generalized bent. By using the knowledge on ideal
decomposition in cyclotomic field, we prove that $f^{**}(x)=f(-x)$ if $f$ is a
generalized bent function and its dual $f^{*}$ is also a generalized bent
function. For any non-weakly regular generalized bent function $f$ which
satisfies that $f(x)=f(-x)$ and its dual $f^{*}$ is generalized bent, we give a
property and as a consequence, we prove that there is no self-dual generalized
bent function $f: V_{n}\rightarrow \mathbb{Z}_{p^k}$ if $p\equiv 3 \ (mod \ 4)$
and $n$ is odd. For $p \equiv 1 \ (mod \ 4)$ or $p\equiv 3 \ (mod \ 4)$ and $n$
is even, we give a secondary construction of self-dual generalized bent
functions. In the end, we characterize the relations between the generalized
bentness of the dual of generalized bent functions and the bentness of the dual
of bent functions, as well as the self-duality relations between generalized
bent functions and bent functions by the decomposition of generalized bent
functions.
