Properties

Degree 2
Conductor $ 3^{2} \cdot 7^{2} \cdot 17^{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  − 4-s + 16-s − 25-s − 2·43-s − 2·47-s + 2·59-s − 64-s − 2·67-s − 2·83-s + 2·89-s + 100-s − 2·101-s − 121-s − 2·127-s − 2·151-s − 169-s + 2·172-s + 2·188-s − 2·236-s + 2·251-s + 256-s − 2·257-s + 2·268-s + 2·293-s + 2·331-s + 2·332-s + 2·353-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127449 ^{s/2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(127449\)    =    \(3^{2} \cdot 7^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((2,\ 127449,\ (1, 1:\ ),\ 1)\)

Euler product

\[\begin{aligned}L(s,\rho) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Particular Values

\[L(1/2,\rho) \approx 0.3751012814\] \[L(1,\rho) \approx 0.6635028172\]

Imaginary part of the first few zeros on the critical line

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