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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).

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).

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).

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).