Properties

Degree 3
Conductor $ 3^{3} \cdot 13 $
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 + 7-s + 8-s + 9-s + 3·11-s + 2·13-s + 17-s + 19-s + 21-s + 23-s + 24-s + 27-s + 29-s + 31-s + 3·33-s + 37-s + 2·39-s + 3·41-s + 2·49-s + 51-s + 53-s + 56-s + 57-s + 63-s + 64-s + 67-s + 69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s) \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(351\)    =    \(3^{3} \cdot 13\)
\( \varepsilon \)  =  $1$
primitive  :  no
self-dual  :  yes
Selberg data  =  \((3,\ 351,\ (0:0),\ 1)\)

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\)\(L(s, \rho_{2.351.3t2.b.a})\)

Particular Values

\[\zeta_K(1/2) \approx -0.6517273998\]
Pole at \(s=1\)

Imaginary part of the first few zeros on the critical line

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