Properties

Degree 3
Conductor $ 3^{5} $
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 + 3-s + 2·4-s + 5-s + 6-s + 2·8-s + 9-s + 10-s + 11-s + 2·12-s + 15-s + 3·16-s + 17-s + 18-s + 2·20-s + 22-s + 23-s + 2·24-s + 2·25-s + 27-s + 29-s + 30-s + 3·32-s + 33-s + 34-s + 2·36-s + 2·40-s + ⋯

Functional equation

\[\begin{aligned} \Lambda_K(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(243\)    =    \(3^{5}\)
\( \varepsilon \)  =  $1$
primitive  :  no
self-dual  :  yes
Selberg data  =  $(3,\ 243,\ (0:0),\ 1)$

Euler product

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

Particular Values

\[\zeta_K(1/2) \approx -1.164313363\]
Pole at \(s=1\)

Imaginary part of the first few zeros on the critical line

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