Properties

Degree 5
Conductor $ 2^{6} \cdot 743 $
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  − 2-s − 5-s + 8-s + 10-s − 13-s − 16-s − 17-s + 26-s + 29-s + 34-s − 37-s − 40-s − 41-s − 43-s + 49-s − 53-s − 58-s − 59-s + 64-s + 65-s + 2·71-s + 74-s + 79-s + 80-s + 82-s + 2·83-s + 85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47552 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1)^{3} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(5\)
\( N \)  =  \(47552\)    =    \(2^{6} \cdot 743\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(5,\ 47552,\ (0, 0, 1, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.1470484491\] \[L(1,\rho) \approx 0.3983297440\]

Imaginary part of the first few zeros on the critical line

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