I saw recently that the $L$-functions of elliptic curves with CM can be factored as a product of simpler $L$-functions. In this question, I'd like to ask why that factorization is significant and what it tells us about the original elliptic curve. Here is the theorem in more detail:
Let $K$ be an imaginary quadratic field, and let $E$ be an elliptic curve with CM by $\mathcal{O}_K$. Suppose that $E$ is defined over a field $L$ containing $K$. Then the $L$-function of $E$ can be factored as follows: $$L(E/L, s) = L(s, \chi_{E/L}) \, L(s, \overline{\chi_{E/L}}) $$ where $\chi_{E/L}$ is the Grossencharacter of $E$ over $L$.
My question is: what is the importance of this factorization? Why is it profound / important? What "underlying truth" about elliptic curves with CM does this theorem reveal?
As an example of what I mean by the last question, here is an example where factorization of $L$-functions has a very nice intuitive meaning: given an abelian number field $K$, you can factor the Dedekind zeta function of $K$ as a product of Dirichlet $L$-functions: $$\zeta_K(s) = \prod_{\chi} L(s, \chi) $$ where $\chi$ runs over all irreducible characters of $\text{Gal }K/\mathbf{Q}$. The "underlying truth" of this theorem is that since characters are periodic functions on the integers, this theorem says that the splitting of primes in abelian extensions is governed by congruence conditions. This equality of $L$-functions is a concise way to encapsulate that fact.
Is there a similar interpretation in the case of CM elliptic curves? What concrete information about the arithmetic of $E$ is captured by the fact that you can split up the $L$-function of $E$ into the pieces $L(s, \chi_{E/L})$ and $L(s, \overline{\chi_{E/L}})$?