Properties

Degree 2
Conductor $ 2^{8} \cdot 3^{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  + 2·23-s − 25-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 + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\R}(s)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2304\)    =    \(2^{8} \cdot 3^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 2304,\ (0, 0:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.7087691341\] \[L(1,\rho) \approx 0.9651586500\]

Imaginary part of the first few zeros on the critical line

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