Properties

Degree 3
Conductor 229
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$\zeta_K(s)$  = 1  + 2-s + 2·4-s + 7-s + 2·8-s + 13-s + 14-s + 3·16-s + 23-s + 26-s + 27-s + 2·28-s + 29-s + 31-s + 3·32-s + 3·37-s + 41-s + 46-s + 47-s + 2·49-s + 2·52-s + 3·53-s + 54-s + 2·56-s + 58-s + 59-s + 62-s + 4·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s)^{3} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(229\)
\( \varepsilon \)  =  $1$
primitive  :  no
self-dual  :  yes
Selberg data  =  \((3,\ 229,\ (0, 0, 0:\ ),\ 1)\)

Euler product

\[\begin{aligned}\zeta_K(s) = \prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\)\(L(s, \rho_{2.229.3t2.a.a})\)

Particular Values

\[\zeta_K(1/2) \approx -0.4290913427\]
Pole at \(s=1\)

Imaginary part of the first few zeros on the critical line

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