Here is the story as I know it. The more general setup is for *induced representations of Langlands type*. These are certain generic representations of $\mathrm{GL}_n(F)$, where $F$ is a nonarchimedean local field. The construction of such representations is the following.

Suppose that $n = n_1 + \cdots + n_r$ with $n_1,\ldots,n_r \geq 1$. Let $\sigma_j$ be a discrete series representation of $\mathrm{GL}_{n_j}(F)$ (necessarily unitary and tempered), from which we can construct a (no longer necessarily unitary or tempered) generic irreducible admissible smooth representation $\pi_j = \sigma_j \otimes \left|\det\right|^{t_j}$ of $\mathrm{GL}_{n_j}(F)$. We form the representation $\pi_1 \otimes \cdots \otimes \pi_r$ of $\mathrm{GL}_{n_1}(F) \times \cdots \times \mathrm{GL}_{n_r}(F)$, which we view as a Levi subgroup in $\mathrm{GL}_n(F)$. Then by normalised parabolic induction, we can induce this to a representation $\pi = \pi_1 \boxplus \cdots \boxplus \pi_r$ of $\mathrm{GL}_n(F)$.

This is known as an induced representation of Whittaker type (in particular, it has a Whittaker model); we say that $\pi$ is the isobaric sum of $\pi_1,\ldots,\pi_r$. If $\Re(t_1) \geq \cdots \geq \Re(t_r)$, then this is said to be an induced representation of Langlands type (which is isomorphic to its Whittaker model). These may not be irreducible, but the irreducible ones are generic irreducible admissible smooth representations, and conversely every generic irreducible admissible smooth representation is isomorphic to an induced representation of Langlands type.

The case you are interested in is $n_1 = \cdots = n_r = 1$, so that each $\pi_j$ is simply a character.

The theorem you are interested in is the following.

The conductor exponent $c(\pi)$ of an induced representation of Langlands type $\pi = \pi_1 \boxplus \cdots \boxplus \pi_r$ is equal to $c(\pi_1) + \cdots + c(\pi_r)$.

Here the conductor exponent is as in my answer here:

On the consistency of the definition of the conductor for automorphic forms

In "Conducteur des représentations du groupe linéaire" by Jacquet, Piatetski-Shapiro, and Shalika, this conductor exponent is defined (see in particular Section 5). In "Rankin-Selberg Convolutions" by the same authors, further properties are shown. In particular, a combination of Theorem 3.1, Proposition 8.4, and Proposition 9.4 shows that the epsilon factor $\varepsilon(s,\pi,\psi)$ satisfies
$$\varepsilon(s,\pi,\psi) = \prod_{j = 1}^{r} \varepsilon(s,\pi_j,\psi),$$
where $\psi$ is an additive character of $F$.

(More precisely, they show the same multiplicativity properties hold for $L(s,\pi)$ and $\gamma(s,\pi,\psi)$, from which the result for $\varepsilon(s,\pi,\psi)$ follows.)

Finally, Théorème 5 of the first paper states that
$$\varepsilon(s,\pi,\psi) = \varepsilon\left(\frac{1}{2},\pi,\psi\right) q^{-c(\pi)\left(s - \frac{1}{2}\right)}$$
(at least for unramified $\psi$), at which point the result follows.