Properties

Degree 3
Conductor $ 3^{2} \cdot 73^{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 − 7-s + 8-s + 9-s − 17-s − 21-s + 24-s + 27-s + 3·43-s + 2·49-s − 51-s − 56-s − 63-s + 64-s + 72-s + 73-s + 81-s − 83-s − 97-s − 103-s + 119-s + 125-s + 3·129-s − 136-s − 137-s − 139-s + 2·147-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 47961 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(47961\)    =    \(3^{2} \cdot 73^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 47961,\ (0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 1.959484684\] \[L(1,\rho) \approx 1.465092591\]

Imaginary part of the first few zeros on the critical line

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