Properties

Degree 3
Conductor $ 5^{2} \cdot 53^{2} $
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  − 0.618·3-s − 5-s − 0.618·7-s + 8-s + 0.999·9-s + 1.61·11-s + 0.618·15-s − 0.618·17-s − 19-s + 0.381·21-s + 1.61·23-s − 0.618·24-s + 25-s − 0.618·29-s − 0.618·31-s − 0.999·33-s + 0.618·35-s − 40-s + 1.61·41-s − 0.999·45-s + 1.61·47-s + 0.999·49-s + 0.381·51-s − 53-s − 1.61·55-s − 0.618·56-s + 0.618·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70225 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(70225\)    =    \(5^{2} \cdot 53^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((3,\ 70225,\ (0, 1, 1:\ ),\ 1)\)

Euler product

\[\begin{aligned}L(s,\rho) = \prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Particular Values

\[L(1/2,\rho) \approx 1.138518236\] \[L(1,\rho) \approx 0.8990109704\]

Imaginary part of the first few zeros on the critical line

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