Properties

Degree 5
Conductor $ 31 \cdot 1433 $
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 − 7-s − 11-s − 13-s − 19-s + 21-s − 23-s + 25-s + 2·27-s + 2·31-s + 32-s + 33-s − 37-s + 39-s − 47-s + 57-s − 61-s + 69-s − 75-s + 77-s + 79-s − 2·81-s − 89-s + 91-s − 2·93-s − 96-s − 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44423 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1)^{3} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(5\)
\( N \)  =  \(44423\)    =    \(31 \cdot 1433\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(5,\ 44423,\ (0, 0, 1, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.1871939049\] \[L(1,\rho) \approx 0.5065393752\]

Imaginary part of the first few zeros on the critical line

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