Let $K$ be a number field and $\mathfrak{m}$ an integral ideal of $\mathcal{O}_K$. Denote by $I(\mathfrak{m})$ the group of fractional ideals prime to $\mathfrak{m}$, and by $P(\mathfrak{m})$ the group of principal ideals generated by an element $\alpha\in K^{\times}$ such that $\alpha\equiv1$ (mod$^{\times}\mathfrak{m}$).
The classical definition of a (unitary) Hecke character that I've seen is as follows: A Hecke character is a continuous homomorphism $I(\mathfrak{m})\to S^1$ for which there exists a character $\chi_{\infty}$ on $(\mathbb{R}^{\times})^{r_1}\times(\mathbb{C}^{\times})^{r_2}$ such that $\chi((\alpha))=\chi_{\infty}^{-1}(1\otimes\alpha)$ for all $(\alpha)\in P(\mathfrak{m})$.
In his book on Modular Forms, Miyake defines a Hecke character to be a continuous homomorphism $\xi:I(\mathfrak{m})\to S^1$ such that $\xi((\alpha))=\prod_{j=1}^{r_1+r_2}(\sigma_j(\alpha)/|\sigma_j(\alpha)|)^{u_j}|\sigma_j(\alpha)|^{iv_j}$ for all $(\alpha)\in P(\mathfrak{m})$, where $\sigma_j$ for $1\leq j\leq r_1$ are the real embeddings, $\sigma_j$ for $r_1\leq j\leq r_1+r_2$ the non-conjugate complex embeddings, $u_j\in\{0,1\}$ for $0\leq j\leq r_1$ and $u_j\in \mathbb{Z}$ otherwise, and $v_j\in\mathbb{R}$ with $\sum_{j}v_j=0$.
Now it's not too hard to determine the dual group of $(\mathbb{R}^{\times})^{r_1} \times(\mathbb{C}^{\times})^{r_2}$, but why is the condition $\sum_{j}v_j=0$ necessary? For instance, if $K=\mathbb{Q}$, $\xi((m))=|m|^{iv}$ seems to be a perfectly valid Hecke character according to the first definition. If this condition is not necessary, then are there any good reasons for imposing it? Any help is greatly appreciated.