Properties

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

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 82369 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(82369\)    =    \(7^{2} \cdot 41^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 82369,\ (0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 1.873234597\] \[L(1,\rho) \approx 1.305369277\]

Imaginary part of the first few zeros on the critical line

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