Properties

Degree $3$
Conductor $105625$
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  + 1.61·2-s + 4-s + 1.61·11-s − 13-s + 2.61·22-s − 1.61·26-s + 27-s + 1.61·29-s − 0.618·31-s + 32-s − 37-s − 0.618·43-s + 1.61·44-s − 0.618·47-s − 52-s + 1.61·53-s + 1.61·54-s + 2.61·58-s − 61-s − 0.999·62-s + 1.61·64-s − 0.618·71-s − 1.61·74-s + 1.61·83-s − 0.999·86-s − 89-s − 0.999·94-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(105625\)    =    \(5^{4} \cdot 13^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 105625,\ (0, 1, 1:\ ),\ 1)\)

Particular Values

\[L(1/2,\rho) \approx 3.666774944\] \[L(1,\rho) \approx 2.342884733\]

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

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