Properties

Degree 3
Conductor $ 7^{2} \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  + 7-s + 8-s − 11-s + 19-s + 27-s − 31-s − 37-s + 49-s + 56-s + 64-s − 77-s − 83-s − 88-s − 103-s − 107-s − 113-s + 2·121-s + 125-s + 133-s − 151-s + 152-s + 3·163-s − 179-s + 189-s − 191-s − 197-s − 209-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(17689\)    =    \(7^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 17689,\ (0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 1.367899544\] \[L(1,\rho) \approx 1.208826868\]

Imaginary part of the first few zeros on the critical line

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