Properties

Degree 3
Conductor $ 2^{6} \cdot 37^{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  + 3·11-s − 23-s + 27-s + 3·29-s − 31-s + 37-s − 43-s − 47-s − 73-s − 97-s − 101-s − 103-s + 6·121-s + 125-s − 137-s − 149-s − 179-s − 191-s − 193-s − 199-s − 211-s − 223-s − 233-s + 3·251-s − 3·253-s − 269-s + 3·297-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 87616 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(87616\)    =    \(2^{6} \cdot 37^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 87616,\ (0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 1.913871716\] \[L(1,\rho) \approx 1.308246425\]

Imaginary part of the first few zeros on the critical line

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