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For an \( n \)-connected graph \( G \), the \( n \)-wide diameter \( d_n(G) \) is the minimum integer \( m \) such that for any two vertices \( x \) and \( y \) there are at least \( n \) internally disjoint paths of length at most \( m \) from \( x \) to \( y \). For a given integer \( l \), a subset \( S \) of \( V(G) \) is called a \( (l,n) \)-dominating set of \( G \) if for any vertex \( x \in V(G) – S \) there are at least \( n \) internally disjoint paths of length at most \( l \) from \( S \) to \( x \). The minimum cardinality among all \( (l,n) \)-dominating sets of \( G \) is called the \( (l,n) \)-domination number. In this paper, we obtain that the \( (l,\omega) \)-domination numbers of the circulant digraph \( G(d^n; \{1, d, \ldots, d^{n-1}\}) \) is equal to 2 for \( 1 \leq \omega \leq n \) and \( d_\omega(G) – (g(d,n) + \delta) \leq l \leq d_\omega(G) – 1 \), where \( g(d,n) = \text{min} \{e\lceil \frac{n}{2} \rceil – e – 2, (\lfloor \frac{n}{2} \rfloor + 1)(e – 1) – 2\} \), \( \delta = 0 \) for \( 1 \leq \omega \leq n – 1 \) and \( \delta = 1 \) for \( \omega = n \).