Consider a Dirichlet series for a non real character of modulus $q$
$$ L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s} $$
and $s\in\Bbb C$ with real part greater than one.
Consider a $J$-series $$ J(s,\chi)=\sum_{n=1}^\infty e^{-\frac{\chi(n)}{n^s}} $$
which converges for $\Re(s)<0.$
Is there a relationship between the analytic continations of $L$ and $J$?
I think that the analytic continuation of $J$ will be partly expressible as a sum of $L-$functions.
I downvoted because your second series doesn't converge. You meant $\chi$ is a Dirichlet character and $$J(s)=\sum_n (e^{-\frac{\chi(n)}{n^s}}-1), \qquad\Re(s) > 1/m$$ where $m$ is the first integer such that $\chi^m$ is principal.
Then expand $$e^{-\frac{\chi(n)}{n^s}}-1=\sum_{k\ge 1} \frac{(-\chi(n))^k n^{-sk}}{k!}$$ and change the order of summation to get a series of Dirichlet L-functions.