Properties

Degree 2
Conductor $ 2^{8} \cdot 17^{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  − 9-s − 25-s − 2·47-s − 49-s + 81-s − 2·89-s − 2·103-s − 121-s − 2·127-s + 2·137-s + 2·151-s − 169-s − 2·191-s + 2·223-s + 225-s − 2·239-s + 2·257-s + 2·263-s − 2·271-s − 2·281-s − 2·353-s − 2·359-s − 361-s − 2·383-s + 2·409-s + 2·423-s − 2·433-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\R}(s)^{2} \, L(s,\rho)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(73984\)    =    \(2^{8} \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((2,\ 73984,\ (0, 0:\ ),\ -1)\)

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0\] \[L(1,\rho) \approx 0.7553996793\]

Imaginary part of the first few zeros on the critical line

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