Properties

Degree 5
Conductor 43063
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  − 3-s − 5-s − 13-s + 15-s − 19-s + 31-s + 32-s − 37-s + 39-s − 41-s + 53-s + 57-s − 59-s + 2·61-s + 65-s − 71-s + 73-s + 3·89-s − 93-s + 95-s − 96-s − 97-s − 109-s + 111-s + 121-s + 123-s − 127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43063 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1)^{3} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(5\)
\( N \)  =  \(43063\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((5,\ 43063,\ (0, 0, 1, 1, 1:\ ),\ 1)\)

Euler product

\[\begin{aligned}L(s,\rho) = \prod_p \ \prod_{j=1}^{5} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Particular Values

\[L(1/2,\rho) \approx 0.1794817505\] \[L(1,\rho) \approx 0.4900062042\]

Imaginary part of the first few zeros on the critical line

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