Properties

Degree $2$
Conductor $57600$
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  + 2·23-s − 2·47-s − 49-s − 2·71-s + 2·73-s + 2·97-s − 121-s − 2·167-s − 169-s − 2·191-s + 2·193-s − 2·239-s + 2·241-s − 2·263-s − 289-s + 2·311-s − 2·313-s − 2·337-s − 2·359-s − 361-s + 2·383-s + 2·409-s + 2·431-s − 2·433-s − 2·457-s + 2·479-s + 2·503-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57600\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 57600,\ (1, 1:\ ),\ -1)\)

Particular Values

\[L(1/2,\rho) \approx 0\] \[L(1,\rho) \approx 0.9869604401\]

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