Properties

Degree $3$
Conductor $614656$
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 + 11-s + (−1.50 + 1.32i)15-s + (−0.499 − 1.32i)23-s − 1.00·25-s + (−0.500 + 1.32i)27-s − 31-s + (−0.499 − 1.32i)33-s + 37-s + (−0.500 + 1.32i)41-s + (0.499 + 1.32i)45-s + (−0.499 − 1.32i)47-s + (−0.499 − 1.32i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ unknown}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(614656\)    =    \(2^{8} \cdot 7^{4}\)
Sign: $unknown$
Primitive: yes
Self-dual: no
Selberg data: \((3,\ 614656,\ (0, 1, 1:\ ),\ 0)\)

Particular Values

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

Euler product

\(L(s,\rho) = \displaystyle\prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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.