Properties

Degree 3
Conductor $ 3^{4} $
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  + 3-s + 8-s + 9-s + 3·17-s + 3·19-s + 24-s + 27-s + 3·37-s + 3·51-s + 3·53-s + 3·57-s + 64-s + 3·71-s + 72-s + 3·73-s + 81-s + 3·89-s + 3·107-s + 3·109-s + 3·111-s + 125-s + 3·127-s + 3·136-s + 3·152-s + 3·153-s + 3·159-s + 3·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s)^{3} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(81\)    =    \(3^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  no
self-dual  :  yes
Selberg data  =  \((3,\ 81,\ (0, 0, 0:\ ),\ 1)\)

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{9}(4, \cdot))\)\(\;\cdot\) \(L(s,\chi_{9}(7, \cdot))\)

Particular Values

\[\zeta_K(1/2) \approx -0.2078465814\]
Pole at \(s=1\)

Imaginary part of the first few zeros on the critical line

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