ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 06 Feb 2021 23:44:49 +0100Dedekind Zeta function of cyclotomic field wrongly evaluating to zero on -1?https://ask.sagemath.org/question/55607/dedekind-zeta-function-of-cyclotomic-field-wrongly-evaluating-to-zero-on-1/Let $K := \mathbb{Q}(\zeta)$ be the pth cyclotomic extension of $\mathbb{Q}$. I would like to verify the results of a paper which states the quotient of their Dedekind zeta functions have particular values. Below Z is the Riemann zeta function (Dedekind zeta function of $\mathbb{Q}$).
x = var('x')
K = NumberField(x**2 + x + 1,'a')
L = K.zeta_function(algorithm='gp')
Z = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
The expected values are nonzero! For example. `(L/Z)(-1)` is expected to be `1.333333333` (i.e. 4/3).
`L(-1)` returns `0.000000000000000`, as does `L(-1)/Z(-1)`.
`Z(-1)` returns `-0.0833333333333333`. `L/Z` returns a type error, as does `L(x)/Z(x)`.
Here is my first question: Have I incorrectly implemented the Dedekind Zeta function of a cyclotomic number field? Why is `L(-1) = 0`?
Here is my second question: How do I implement the evaluation of the L-series after I've taken their quotient? That is, `A = L/Z, A(-1);` instead of `L(-1)/Z(-1)`. tzeentchSat, 06 Feb 2021 23:44:49 +0100https://ask.sagemath.org/question/55607/