"Mollifier" of the Dirichlet L-function

Question

I was studying some zero-density results for $\zeta(s)$, mostly from Titchmarsh's book "The Theory of the Riemann zeta function", Chapter 9. In one place, as per the literature, a mollifier of $\zeta(s)$ is defined as $M_{X}(s) = \sum_{n\leq X}\frac{\mu(n)}{n^s}$, and subsequently a function $f_{X}(s)$ defined as $f_{X}(s) = \zeta(s)M_{X}(s) -1 = \sum_{m\geq 1}\frac{1}{m^s}\sum_{n\leq X}\frac{\mu(n)}{n^s} -1 = \sum_{n > X} \frac{\sum_{d\vert n, d\leq X}\mu(d)}{n^s} =\sum_{n\geq 1}\frac{a_{X}(n)}{n^s}$, being the series representation of $f_{X}(s)$, where $a_{X}(n) = 0$ if $n\leq X$ and $a_{X}(n) = \sum_{d\vert n, d \leq X} \mu(d)$ if $n>X$. My question is if I define a mollifier in similar lines for the Dirichlet $L$-function as $M_{X}(s, \chi) = \sum_{n\leq X} \frac{\mu(n)\chi(n)}{n^s}$ and subsequently define $f_{X}(s, \chi) = M_{X}(s,\chi)L(s, \chi) - 1$, what can be the series representation for $f_{X}(s, \chi)$? What can be the expression of the general term?