# How to define a function the Dirichlet L-function $L(s,\overline{\chi(5,2)})$ in Mathematica?

In Mathematica:

The Dirichlet L-function with character $\chi(5,2)$, $L(s,\chi(5,2))$, is expressed as DirichletL[5,2,s]

Let $\overline{\chi(5,2)}$ be the complex conjugate of $\chi(5,2)$. How to define the function $L(s,\overline{\chi(5,2)})$ in Mathematica?

Thanks- mike

Maybe by Conjugate[DirichletL[5, 2, Conjugate[s]]]?
$$L(\overline{\chi(5,2)}, s) = \sum_{n=1}^\infty \overline{\chi(5,2)(n)} \, n^{-s} = \sum_{n=1}^\infty \overline{ \chi(5,2)(n) \, \overline{n^{-s}}}$$
and since $n$ is real, we have $\overline{n^{-s}} = n^{-\overline{s}}$.
• Thanks a lot! It turned out that $\overline{\chi(5,2)(n)}= \chi(5,4)(n)$. So this is workaround. But your answer is definitely much better. – mike Apr 29 '18 at 11:27