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Let $$D$$ be a simple digraph (directed graph) with vertex set $V(D)$ and arc set $A(D)$ where $n=|V(D)|$, and each arc is an ordered pair of distinct vertices. If $$(v,u) \in A(D)$$, then $$u$$ is considered an \emph{out-neighbor} of $$v$$ in $$D$$. Initially, we designate each vertex to be either filled or empty. Then, the following color change rule (CCR) is applied: if a filled vertex $$v$$ has exactly one empty out-neighbor $$u$$, then $$u$$ will be filled. If all vertices in $V(D)$$ are eventually filled under repeated applications of the CCR, then the initial set is called a \emph{zero forcing set} (ZFS); if not, it is a \emph{failed zero forcing set} (FZFS). We introduce the \emph{failed zero forcing number} $$\F(D)$$ on a digraph, which is the maximum cardinality of any FZFS. The \emph{zero forcing number}, $$\Z(D)$, is the minimum cardinality of any ZFS. We characterize digraphs that have $$\F(D)<\Z(D)$$ and determine $$\F(D)$$ for several classes of digraphs including de Bruijn and Kautz digraphs. We also characterize digraphs with $$\F(D)=n-1$$, $$\F(D)=n-2$$, and $$\F(D)=0$$, which leads to a characterization of digraphs in which any vertex is a ZFS. Finally, we show that for any integer $$n \geq 3$$ and any non-negative integer $$k$$ with $k