Properties

Degree $6$
Conductor $237751$
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 − 3-s − 5-s + 6-s − 7-s + 9-s + 10-s − 11-s + 14-s + 15-s − 17-s − 18-s + 21-s + 22-s − 27-s + 29-s − 30-s − 31-s + 33-s + 34-s + 35-s + 2·41-s − 42-s − 45-s + 47-s + 51-s − 53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 237751 ^{s/2} \, \Gamma_{\R}(s)^{3} \, \Gamma_{\R}(s+1)^{3} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(237751\)    =    \(23 \cdot 10337\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((6,\ 237751,\ (0, 0, 0, 1, 1, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 0.04679975965\] \[L(1,\rho) \approx 0.2314834234\]

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

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