Properties

Degree 3
Conductor $ 7^{2} \cdot 43 $
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  − 7-s − 8-s − 13-s − 27-s + 29-s − 41-s + 2·43-s + 49-s + 56-s + 64-s − 3·71-s − 83-s + 91-s + 3·97-s + 104-s + 113-s − 125-s − 127-s − 139-s + 3·167-s + 2·169-s − 181-s + 189-s − 197-s − 203-s − 3·211-s + 216-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(2107\)    =    \(7^{2} \cdot 43\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((3,\ 2107,\ (0, 0, 1:\ ),\ 1)\)

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.3406582022\] \[L(1,\rho) \approx 0.7085138813\]

Imaginary part of the first few zeros on the critical line

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