Properties

Degree 4
Conductor $ 5^{2} \cdot 11 \cdot 29^{3} $
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  + 11-s − 16-s − 25-s − 29-s + 2·61-s − 2·79-s + 81-s − 2·109-s − 2·131-s + 2·139-s + 2·149-s − 176-s + 2·229-s + 2·251-s + 256-s + 2·269-s − 275-s + 2·281-s + 2·311-s − 319-s + 2·361-s + 2·379-s − 2·389-s + 400-s + 2·409-s − 2·431-s + 2·461-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6706975 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{3} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(6706975\)    =    \(5^{2} \cdot 11 \cdot 29^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((4,\ 6706975,\ (0, 1, 1, 1:\ ),\ 1)\)

Euler product

\[\begin{aligned}L(s,\rho) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Particular Values

\[L(1/2,\rho) \approx 1.520655642\] \[L(1,\rho) \approx 1.018985533\]

Imaginary part of the first few zeros on the critical line

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