Approximate functional equation for the product of two Dirichlet $L$-functions

An approximate functional is derived for $L(s,\chi_1)L(s,\chi_2)$, where $\chi_1$ and $\chi_2$ are primitive Dirichlet characters modulo $k_1$ and $k_2$, and also an approximate functional equation for an analogue of the Hardy–Selberg function.
If $s=1/2+it$, $k_1k_2\leqslant |t|^{1/9 -5\varepsilon}$, then the remainder terms in these formulas are bounded by $O(|t|^{-\varepsilon})$ as $|t|\to\infty$ (where $\varepsilon$ is an arbitrarily small positive number).