Properties

Degree 3
Conductor $ 2^{6} \cdot 5^{4} $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  − 0.618·3-s − 5-s − 0.618·7-s + 0.999·9-s − 0.618·11-s + 0.618·15-s + 1.61·19-s + 0.381·21-s − 0.618·23-s + 25-s − 29-s − 31-s + 0.381·33-s + 0.618·35-s − 41-s − 43-s − 0.999·45-s + 1.61·47-s + 0.999·49-s − 53-s + 0.618·55-s − 0.999·57-s + 1.61·61-s − 0.618·63-s + 0.381·69-s + 1.61·71-s − 0.618·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(40000\)    =    \(2^{6} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((3,\ 40000,\ (0, 1, 1:\ ),\ 1)\)

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.7248521373\] \[L(1,\rho) \approx 0.7047713309\]

Imaginary part of the first few zeros on the critical line

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