Properties

Degree 7
Conductor $ 313^{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  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 11-s + 12-s + 13-s + 16-s − 17-s − 18-s + 22-s − 24-s − 26-s + 3·27-s − 29-s − 31-s − 32-s − 33-s + 34-s + 36-s − 37-s + 39-s + 41-s + 43-s − 44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(313^{4}\right)^{s/2} \, \Gamma_{\R}(s)^{3} \, \Gamma_{\R}(s+1)^{4} \, L(s,\rho)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(7\)
\( N \)  =  \(313^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(7,\ 313^{4} ,\ ( 0, 0, 0, 1, 1, 1, 1 : \ ),\ 1 )$

Euler product

\[\begin{aligned}L(s,\rho) = \prod_p \ \prod_{j=1}^{7} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Particular Values

\[L(1/2,\rho) \approx 0.8741060067\] \[L(1,\rho) \approx 0.9413236133\]

Imaginary part of the first few zeros on the critical line

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