Properties

Degree $2$
Conductor $112896$
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  + 2·11-s − 2·13-s − 25-s + 2·47-s + 2·61-s − 2·107-s + 3·121-s − 4·143-s − 2·157-s + 2·167-s + 3·169-s − 2·179-s + 2·181-s + 2·193-s + 2·229-s − 2·275-s − 289-s + 2·311-s + 2·325-s − 2·337-s − 2·347-s + 2·349-s − 361-s + 2·383-s + 2·397-s − 2·443-s + 2·457-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\R}(s)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112896\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 112896,\ (0, 0:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 1.342179866\] \[L(1,\rho) \approx 1.074298105\]

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