Properties

Degree 2
Conductor $ 2^{8} \cdot 3^{2} \cdot 7^{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  − 2·5-s + 3·25-s − 2·43-s − 2·47-s + 2·67-s − 2·101-s − 121-s − 4·125-s + 2·163-s + 2·167-s − 169-s − 2·173-s + 2·193-s − 2·211-s + 4·215-s + 4·235-s + 2·269-s − 289-s + 2·293-s − 2·311-s + 2·331-s − 4·335-s + 2·337-s − 361-s − 2·379-s − 2·383-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

\( d \)  =  \(2\)
\( N \)  =  \(112896\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 112896,\ (0, 0:\ ),\ -1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0\] \[L(1,\rho) \approx 0.6181346098\]

Imaginary part of the first few zeros on the critical line

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