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  + 2·7-s − 25-s + 3·49-s + 2·79-s − 2·89-s + 2·97-s − 2·113-s − 2·127-s + 2·137-s − 2·151-s − 2·167-s − 169-s − 2·175-s + 2·239-s + 2·257-s + 2·263-s + 2·271-s − 289-s + 2·313-s + 4·343-s − 2·353-s + 2·359-s − 361-s + 2·401-s + 2·431-s − 2·433-s + 2·439-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

\[L(1/2,\rho) \approx 3.577892883\] \[L(1,\rho) \approx 1.345855354\]

Imaginary part of the first few zeros on the critical line

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