Properties

Degree 3
Conductor $ 2^{3} \cdot 127 $
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 + 4-s − 12-s − 13-s + 16-s − 29-s − 31-s − 37-s + 39-s − 43-s + 47-s − 48-s − 52-s + 59-s + 61-s + 64-s + 67-s + 71-s − 73-s + 79-s + 81-s + 83-s + 87-s − 89-s + 93-s + 97-s − 103-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1016 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(1016\)    =    \(2^{3} \cdot 127\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 1016,\ (0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.5290640284\] \[L(1,\rho) \approx 0.7891312894\]

Imaginary part of the first few zeros on the critical line

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