Zeta-type functional equation conjecture for the exponential congruence symbol

Let $n$ and $k$ be fixed, let $\chi_k$ be the exponential congruence symbol, and define its Dirichlet series by

$$L(s,\chi_k)=\sum_{m=1}^{\infty}\frac{\chi_k(m)}{m^s}.$$

Define the completed function

$$\Lambda(s,\chi_k)=\pi^{-s/2}\Gamma\!\left(\tfrac{s}{2}\right)L(s,\chi_k).$$

Zeta-type relation. The completed function may satisfy a functional equation of the form

$$\Lambda(s,\chi_k)=W\cdot\Lambda(1-s,\chi_k),$$

with a constant $W$ depending on $n$ and $k$.

This conjecture seeks an analogue of the functional equation for completed classical $L$-functions, but it remains open for the exponential congruence symbol.