Properties

Degree 3
Conductor $ 2^{3} \cdot 971^{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  + (−0.500 + 1.32i)3-s + 7-s − 1.00·9-s + (−0.499 − 1.32i)13-s + (−0.500 + 1.32i)17-s + (−0.499 − 1.32i)19-s + (−0.500 + 1.32i)21-s − 23-s + (−0.499 − 1.32i)27-s − 29-s + (−0.500 + 1.32i)31-s + (−0.500 + 1.32i)37-s + 2·39-s + 41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7542728 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ unknown}) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(7542728\)    =    \(2^{3} \cdot 971^{2}\)
\( \varepsilon \)  =  $unknown$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(3,\ 7542728,\ (0, 1, 1:\ ),\ 0)$

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.