Properties

Degree $3$
Conductor $327$
Sign $1$
Motivic weight $0$
Arithmetic yes
Primitive no
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$\zeta_K(s)$  = 1  + 2·3-s + 5-s + 8-s + 3·9-s + 3·11-s + 13-s + 2·15-s + 19-s + 2·24-s + 2·25-s + 4·27-s + 29-s + 6·33-s + 37-s + 2·39-s + 40-s + 3·45-s + 3·53-s + 3·55-s + 2·57-s + 64-s + 65-s + 67-s + 71-s + 3·72-s + 4·75-s + 79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 327 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(327\)    =    \(3 \cdot 109\)
Sign: $1$
Arithmetic: yes
Primitive: no
Self-dual: yes
Selberg data: \((3,\ 327,\ (0:0),\ 1)\)

Particular Values

\[\zeta_K(1/2) \approx -0.9015133868\]
Pole at \(s=1\)

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\)\(L(s, \rho_{2.327.3t2.a.a})\)

Imaginary part of the first few zeros on the critical line

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