Properties

Degree 3
Conductor $ 163^{2} $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  − 5-s + 8-s − 13-s − 17-s − 23-s + 2·25-s + 27-s − 31-s − 37-s − 40-s − 53-s − 59-s − 61-s + 64-s + 65-s + 85-s − 104-s + 115-s − 2·125-s − 127-s − 135-s − 136-s + 155-s − 157-s + 163-s + 2·169-s − 184-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 26569 ^{s/2} \, \Gamma_{\R}(s)^{3} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(26569\)    =    \(163^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 26569,\ (0, 0, 0:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.3724739457\] \[L(1,\rho) \approx 0.7710042337\]

Imaginary part of the first few zeros on the critical line

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