Properties

Degree 3
Conductor $ 3^{4} \cdot 19^{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  + 1.61·2-s + 4-s + 1.61·5-s − 7-s + 2.61·10-s + 1.61·11-s − 1.61·14-s − 19-s + 1.61·20-s + 2.61·22-s − 0.618·23-s + 25-s − 28-s − 0.618·29-s + 32-s − 1.61·35-s − 1.61·38-s − 41-s + 1.61·44-s − 0.999·46-s − 47-s + 2·49-s + 1.61·50-s + 2.61·55-s − 0.999·58-s + 1.61·61-s + 1.61·64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(29241\)    =    \(3^{4} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 29241,\ (0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 3.373615240\] \[L(1,\rho) \approx 2.545096405\]

Imaginary part of the first few zeros on the critical line

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