Properties

Degree 2
Conductor $ 2^{8} \cdot 3^{2} \cdot 11^{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  + 2·5-s − 2·19-s + 3·25-s + 2·43-s − 49-s − 2·53-s − 4·95-s + 2·97-s + 4·125-s − 2·139-s + 2·167-s − 169-s − 2·211-s + 4·215-s − 2·239-s − 2·245-s − 2·263-s − 4·265-s + 2·269-s − 2·283-s − 289-s + 2·307-s − 2·313-s + 2·317-s + 2·359-s + 3·361-s − 2·389-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(278784\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((2,\ 278784,\ (1, 1:\ ),\ -1)\)

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0\] \[L(1,\rho) \approx 1.345855145\]

Imaginary part of the first few zeros on the critical line

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