The L-function of CM elliptic curve $E$ over an imaginary quadratic field can be written as the product of the Hecke L-functions (for simplicity, I assume that the base field of the elliptic curve and the CM field is the same field $K$). This Hecke character's target is $\mathbb{C}^{*}$ and not $S^1$.

The Hecke character is often written as a character whose values are in the unit circle. If we write

$L(E,s):=L(\chi,s)L(\overline{\chi},s)$ $\quad$ ($\chi$ is the Hecke character whose values are not in $S^1$)

$\mathbb{L}(E,s):=L(\chi',s)L(\overline{\chi'},s)$ $\quad$ ($\chi'$ is the normalize Hecke character whose values are in $S^1$),

my question is if one can relate $L(E,1)$ and $\mathbb{L}(E,1)$. Please tell me.