# Hecke character and CM elliptic curve

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.

• Do you know how to express the $L$-function of a non-normalized Hecke character $\chi$ in terms of the $L$-function of its normalized form $\chi'$? – KConrad Jan 3 '16 at 15:44

The two $L$-functions are shifts of each other. That is, $\mathbb{L}(E,s)=L(E,s+s_0)$ for some fixed $s_0$. BTW some people do not appreciate the importance to shift normalize every automorphic $L$-function so that its center is $s=1/2$. This normalization creates order in the world and pays respect to Riemann as well.
• Indeed, specifically, for elliptic curves there is a "competing" tradition of making the critical strip $0\le \sigma\le 2$, and critical line $\sigma=1$... – paul garrett Jan 3 '16 at 17:41
• It makes sense to use the other normalization for algebraicity issues; for instance, the field generated over $\mathbf{Q}$ by the coefficients of the Hecke-normalized $L$-function of an elliptic curve (i.e., functional equation between $s$ and $1-s$) is of infinite degree over $\mathbf{Q}$, and depends on the elliptic curve (through the set of supersingular primes, for instance). – Denis Chaperon de Lauzières Jan 3 '16 at 19:05
• @DenisChaperondeLauzières: No doubt it makes sense, but I don't like it, especially that most automorphic $L$-functions seem highly transcendental. Algebraicity is a rare feature. – GH from MO Jan 3 '16 at 19:27
• Somewhat related to @DenisChaperondeLauzières's comment, although I myself do like the $s\leftrightarrow 1-s$ convention, it is true that Deligne's conjectures, e.g., proven in some cases by Shimura and others for holomorphic elliptic or Hilbert modular forms' L-functions, about special values, do seem to "want" a wider critical strip, etc. And, yes, these results are highly anomalous. Once again, I think there's no single best convention... so one always must think about context. In a different universe things might have been otherwise... – paul garrett Jan 3 '16 at 22:53
• @paulgarrett: I think the symbol $L(s,\pi)$ should be reserved for the finite $L$-function $\prod_p L_p(s,\pi)$ with functional equation $s\leftrightarrow 1-s$ and $\pi\leftrightarrow\tilde\pi$. Anything else, e.g. $L(s,\pi)$ with the gamma factors attached, or $L(s,\pi)$ shifted by some constant or rescaled in any way, should be denoted with a different symbol such as $\Lambda(s,\pi)$ or $D(s,\pi)$. This could be carried out in our universe. – GH from MO Jan 4 '16 at 1:48