# 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)`

.