Properties

Degree 3
Conductor $ 2^{4} \cdot 53^{2} $
Sign $unknown$
Motivic weight 0
Primitive yes
Self-dual no

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + (−0.499 − 1.32i)3-s + (−0.499 − 1.32i)5-s − 1.00·9-s + (−0.500 + 1.32i)11-s + (−0.500 + 1.32i)13-s + (−1.50 + 1.32i)15-s + (−0.500 + 1.32i)17-s − 1.00·25-s + (−0.500 + 1.32i)27-s + 2·33-s + 2·39-s + 41-s − 43-s + (0.499 + 1.32i)45-s − 47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44944 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ unknown}) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(44944\)    =    \(2^{4} \cdot 53^{2}\)
\( \varepsilon \)  =  $unknown$
primitive  :  yes
self-dual  :  no
Selberg data  =  \((3,\ 44944,\ (0, 1, 1:\ ),\ 0)\)

Euler product

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

Particular Values

Not enough information (Dirichlet series coefficients/sign of the functional equation) to compute special values.

Imaginary part of the first few zeros on the critical line

Zeros not available.

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

Plot not available.