Properties

Degree 3
Conductor $ 2^{6} \cdot 31^{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-s + 4-s − 8-s + 16-s − 23-s + 27-s − 29-s + 31-s − 32-s + 46-s − 47-s − 54-s + 58-s − 61-s − 62-s + 64-s − 89-s − 92-s + 94-s − 97-s − 101-s + 108-s + 3·109-s − 116-s + 122-s + 124-s + 125-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 61504 ^{s/2} \, \Gamma_{\R}(s)^{3} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(61504\)    =    \(2^{6} \cdot 31^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 61504,\ (0, 0, 0:\ ),\ 1)$

Euler product

\[\begin{aligned} L(s,\rho) = \prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Particular Values

\[L(1/2,\rho) \approx 0.3399558778\] \[L(1,\rho) \approx 0.6202901588\]

Imaginary part of the first few zeros on the critical line

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