Properties

Degree 3
Conductor $ 2^{6} \cdot 5^{4} $
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  + 5-s + 1.61·7-s − 11-s + 1.61·13-s − 17-s − 0.618·19-s + 25-s + 27-s − 0.618·29-s + 1.61·35-s − 41-s − 0.618·43-s + 1.61·47-s + 49-s − 55-s − 0.618·59-s + 1.61·65-s − 67-s + 1.61·71-s − 1.61·77-s − 0.618·79-s − 85-s − 89-s + 2.61·91-s − 0.618·95-s + 1.61·97-s − 0.618·103-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

\[L(1/2,\rho) \approx 1.917045898\] \[L(1,\rho) \approx 1.441299733\]

Imaginary part of the first few zeros on the critical line

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