Properties

Degree 3
Conductor $ 2^{6} \cdot 3^{2} \cdot 11^{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  − 3-s + 1.61·5-s + 9-s + 11-s − 1.61·15-s − 0.618·17-s + 1.61·19-s + 25-s − 27-s − 29-s − 0.618·31-s − 33-s + 1.61·45-s − 0.618·47-s + 0.618·51-s − 0.618·53-s + 1.61·55-s − 1.61·57-s + 1.61·59-s + 1.61·61-s + 1.61·67-s + 1.61·71-s − 73-s − 75-s + 81-s − 0.999·85-s + 87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(69696\)    =    \(2^{6} \cdot 3^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((3,\ 69696,\ (0, 1, 1:\ ),\ 1)\)

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 1.577089122\] \[L(1,\rho) \approx 1.144130706\]

Imaginary part of the first few zeros on the critical line

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