Properties

Degree $3$
Conductor $239$
Sign $1$
Motivic weight $0$
Arithmetic yes
Primitive no
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$\zeta_K(s)$  = 1  + 3·3-s + 7-s + 8-s + 6·9-s + 13-s + 19-s + 3·21-s + 23-s + 3·24-s + 10·27-s + 37-s + 3·39-s + 41-s + 43-s + 47-s + 2·49-s + 53-s + 56-s + 3·57-s + 59-s + 6·63-s + 64-s + 3·69-s + 3·71-s + 6·72-s + 73-s + 79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 239 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(239\)
Sign: $1$
Arithmetic: yes
Primitive: no
Self-dual: yes
Selberg data: \((3,\ 239,\ (0:0),\ 1)\)

Particular Values

\[\zeta_K(1/2) \approx -0.9633902223\]
Pole at \(s=1\)

Euler product

\(\zeta_K(s) = \displaystyle\prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\)\(L(s, \rho_{2.239.3t2.a.a})\)

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line