Properties

Degree 3
Conductor $ 13^{2} $
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  + 3·5-s + 8-s + 13-s + 6·25-s + 27-s + 3·31-s + 3·40-s + 3·47-s + 3·53-s + 64-s + 3·65-s + 3·73-s + 3·79-s + 3·83-s + 3·103-s + 104-s + 3·109-s + 10·125-s + 3·131-s + 3·135-s + 3·151-s + 9·155-s + 3·157-s + 169-s + 3·181-s + 6·200-s + 216-s + ⋯

Functional equation

\[\begin{aligned}\Lambda_K(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s)^{3} \, \zeta_K(s)\cr =\mathstrut & \, \Lambda_K(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(169\)    =    \(13^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  no
self-dual  :  yes
Selberg data  =  \((3,\ 169,\ (0, 0, 0:\ ),\ 1)\)

Euler product

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

Factorization

\(\zeta_K(s) =\) \(\zeta(s)\)\(\;\cdot\) \(L(s,\chi_{13}(3, \cdot))\)\(\;\cdot\) \(L(s,\chi_{13}(9, \cdot))\)

Particular Values

\[\zeta_K(1/2) \approx -0.2674316037\]
Pole at \(s=1\)

Imaginary part of the first few zeros on the critical line

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