Properties

Degree $2$
Conductor $278784$
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  − 2·17-s − 25-s − 2·31-s + 2·41-s − 49-s − 2·97-s − 2·103-s − 2·167-s − 169-s − 2·199-s − 2·223-s + 2·233-s + 2·239-s + 2·263-s − 2·281-s + 3·289-s − 2·313-s − 2·359-s − 361-s − 2·367-s + 2·425-s − 2·431-s − 2·433-s + 2·463-s − 2·479-s − 2·487-s − 2·503-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

Degree: \(2\)
Conductor: \(278784\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 278784,\ (1, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 0.1840328266\] \[L(1,\rho) \approx 0.7476976378\]

Euler product

\(L(s,\rho) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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