Properties

Degree $6$
Conductor $9037905787$
Sign $unknown$
Motivic weight $0$
Arithmetic yes
Primitive yes
Self-dual no

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + i·2-s + i·3-s − 4-s i·5-s − 6-s i·7-s i·8-s − 9-s + 10-s i·12-s + 14-s + 15-s + 16-s + 17-s i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2083^{3}\right)^{s/2} \, \Gamma_{\R}(s)^{3} \, \Gamma_{\R}(s+1)^{3} \, L(s,\rho)\cr=\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})}\quad (\text{with }\epsilon \text{ unknown})\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2083^{3}\)
Sign: $unknown$
Arithmetic: yes
Primitive: yes
Self-dual: no
Selberg data: \((6,\ 2083^{3} ,\ ( 0, 0, 0, 1, 1, 1 : \ ),\ 0 )\)

Particular Values

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

Euler product

\(L(s,\rho) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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.