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I need to do some numerical computation on special values of a Hecke L-function $L(s,\chi)$. To do this, I want to construct a Hecke character in MAGMA, given that I know its infinity type.

In other words, suppose we are working on a totally real field $K$. My Hecke character $\chi$ is first defined over the principal ideals by

$$\chi((\alpha))=\prod sgn(\sigma_i(\alpha))^{m_i}|\sigma_i(\alpha)|^{n_i}$$

where $\sigma_i$'s are the real embeddings, $m_i=0$ or $1$, and $n_i\in \mathbb{C}$. For some good values of $m,n$ this can be extended to all ideals, thus becomes a Hecke character.

I tried to read https://magma.maths.usyd.edu.au/magma/handbook/text/410 for related information. It looks like the functions related to Hecke Grössencharacters are close to what I need, but it requires CM field to work, while I want to deal, say with $K=\mathbb{Q}(\sqrt{3})$.

Edit1: Thanks to Jeremy Rouse, I read http://magma.maths.usyd.edu.au/magma/handbook/text/1485 about creating a general L-series in Magma. It assumes that the shifts in the gamma factors are rational, while I want to do the following example:

$K=\mathbb{Q}(\sqrt{3})$, $\chi(\alpha)=Sgn(\alpha\alpha')(\alpha/\alpha')^{i\pi/R}$ where $R$ is the regulator (so that $\chi$ is $1$ on units).

$L(\chi)=\sum_{I\subset O_K}\chi(I)N(I)^{-s}$, whose gamma factor is

$$\nu(s)=(dπ^{−2})^{s/2}Γ(s/2+1/2+πi/2R)Γ(s/2+1/2−πi/2R).$$

I start with the following Magma code:

K:=QuadraticField(3);
C<i>:=ComplexField();
pi:=Pi(C);
d:=AbsoluteDiscriminant(K); d;
r:=Regulator(K);
mu:=pi/2/r; mu;
L := LSeries(1, [mu,-mu], d, 0: Sign:=1);
N := LCfRequired(L); N;

which results in "Runtime error in 'LSeries': elements of gamma must be integer or rational numbers". (So as Jeremy commented later, this cannot be doen with Magma. with add some detail if I work it out.)

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  • $\begingroup$ I am skeptical that the $L$-function properly has "special values", as it is not motivic. $\endgroup$ – kantelope Dec 8 '15 at 22:46
  • $\begingroup$ @kantelope I should say that I want to evaluate it at some integer/half integer. $\endgroup$ – Ted Mao Dec 8 '15 at 23:11
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It sounds like Magma does not currently include the exact functionality that you want to use. An alternative that you could try is manually building the $L$-function you wish to evaluate numerically (see the section of the Magma documentation "Arithmetic Geometry - $L$-functions - Constructing a general L-series"). This accesses the algorithms (primarily based on work of Tim Dokchitser) for computing $L$-functions. I have manually built and evaluated many $L$-functions (including for example, symmetric power $L$-functions of modular forms).

In order to do this, you need to know the Euler factors, line of symmetry, conductor, and Gamma factors. (Magma will guess the sign of the functional equation for you.) This information was worked out by Hecke - an excellent modern reference is Iwaniec's "Topics in Classical Automorphic Forms" - Section 12.2.

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  • $\begingroup$ Thanks for the information. However, I am still confused about my case. The "General L-series" Magma page has an assumption on the gamma factor $\nu(s)=\Gamma(s+\lambda_1 /2)\ldots\Gamma(s+\lambda_d /2)$. Now we just have real places and there will be $\Gamma(s/2)$ type factors only. Does it still work? $\endgroup$ – Ted Mao Dec 8 '15 at 18:23
  • $\begingroup$ Or, can you give me an example, say $\chi(\alpha)=Sgn(\alpha\bar{\alpha}) (\alpha\bar{\alpha}^{-1})^{i\pi/R}$ where $R$ is the regulator (so $\chi$ is 1 on the units), and $L(s)=\sum_{I\subset O_K} \chi(I) N(I)^{-s}$, where $K=\mathbb{Q}(\sqrt{3})$. In this case $\nu(s)=(d\pi^{-2})^{s/2} \Gamma(s/2+1/2+i\pi/2R)\Gamma(s/2+1/2-i\pi/2R)$ $\endgroup$ – Ted Mao Dec 8 '15 at 18:31
  • $\begingroup$ So, after playing with this for a while, it turns out you cannot do this in Magma. The form of the gamma factors in the Magma documentation is a typo - instead they are expected to be $\Gamma\left(\frac{s+\lambda_{1}}{2}\right) \cdots$. However, Magma expects the $\lambda_{i}$ to be rational, and this is a problem in your example! This is a factor of the Magma implementation - if you use Tim Dokchitser's original PARI script, you'll be fine. Another possibility is Michael Rubinstein's "lcalc". $\endgroup$ – Jeremy Rouse Dec 8 '15 at 20:05

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