In his thesis Tate deduces a functional equation for Hecke L-Functions associated to a unitary idele class character $\chi$ of the form $$ L(s,\chi)=\epsilon(s,\chi)L(1-s,\overline{\chi}).$$ Here the global $\epsilon$-factor admits a factorization into its local components at a place $v$ $$\epsilon(s,\chi)=\prod_v \epsilon_v(s,\chi_v).$$ where $\chi_v$ is the component of $\chi$ at the place $v$. The local root number now is defined as $W(\chi_v)=\epsilon_v\left(\frac12,\chi_v\right)$.
I have as an exercise to show that the local root number is a root of unity if the exponent of the conductor of $\chi_v$ is not $1$. Does anybody knows a proof or some helpful facts about this?
What I already know is that in the case where the conductor is $0$ one can express $\chi_v$ as $\chi_v=|\cdot|_v^s$ for complex $s$ and $|\cdot|_v$ is the normalized value at the place $v$. The root number can now be expressed as $$W(\chi_v)=|\pi^d|^s$$ where $\pi$ is the local uniformizing parameter and $d$ is the exponent of the local different. Why is this a root of unity?
The idea behind proving the root number (under suitable conditions) is a root of unity is to appeal to the following result: an algebraic number $\alpha$ in a number field $K$ is a root of unity iff $|\alpha|_v = 1$ for all places $v$ of $K$. Note that if $|\alpha|_v = 1$ for all non-Archimedean places $v$ of $K$ then $\alpha \in \mathcal O_K$, so we're now reduced to an old result of Kronecker, which says an algebraic integer $\alpha$ is a root of unity iff $|\alpha|_v = 1$ for all archimedean places $v$, i.e., $|\sigma(\alpha)| = 1$ for all archimedean embeddings $\sigma$ of $K$ into $\mathbf R$ and $\mathbf C$.
For a Dirichlet character with prime power conductor $p^k > 1$, you're saying its root number is a root of unity if $k \geq 2$. To appreciate this statement, I'll tell you what happens if $k = 1$ (exponent $1$ in your terminology).
If $\chi = (\frac{\cdot}{p})$ is the Legendre symbol modulo a prime $p$ then its root number is $1$ (a trivial root of unity), but if $\chi$ is a nontrivial Dirichlet character mod $p$ that is not the Legendre symbol (so it is a nonquadratic character) then the root number $W(\chi)$ is not a root of unity. That is due independently to Chowla and Mordell. See S. Chowla, On Gaussian sums, Proc. National Acad. Sci. USA 48 (1962), 1127-1128 and L. J. Mordell, On a cyclotomic resolvent, Arch. Math. (Basel) 13 (1962), 486-487. Mordell writes at the start of his paper that Chowla had conjectured the root number is not a root of unity for nontrivial characters mod $p$ other than the Legendre symbol.
For example, if $\chi \bmod 5$ is the quartic Dirichlet character where $\chi(2) = i$ then $W(\chi)^4 = -3/5 + (4/5)i$, which is not a root of unity, so $W(\chi)$ is not a root of unity.