Generalized Turán problem for directed cycles

For integers $k,\mathrm{\ell }\ge 3$, let $\mathrm{e}\mathrm{x}(n,\overrightarrow{C_k},\overrightarrow{C_{\mathrm{\ell }}})$ denote the maximum number of directed cycles of length $k$ in any oriented graph on $n$ vertices which does not contain a directed cycle of length $\mathrm{\ell }$. We establish the order of magnitude of $\mathrm{e}\mathrm{x}(n,\overrightarrow{C_k},\overrightarrow{C_{\mathrm{\ell }}})$ for every $k$ and $\mathrm{\ell }$ and determine its value up to a lower error term when $k\nmid \mathrm{\ell }$ and $\mathrm{\ell }$ is large enough. Additionally, we calculate the value of $\mathrm{e}\mathrm{x}(n,\overrightarrow{C_k},\overrightarrow{C_{\mathrm{\ell }}})$ for some other specific pairs $(k,\mathrm{\ell })$ showing that a diverse class of extremal constructions can appear for small values of $\mathrm{\ell }$.