Properties

Degree 3
Conductor $ 11 \cdot 233 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  − 2-s − 3-s − 5-s + 6-s + 7-s + 10-s + 2·11-s + 13-s − 14-s + 15-s + 16-s − 19-s − 21-s − 2·22-s − 26-s − 29-s − 30-s − 32-s − 2·33-s − 35-s + 38-s − 39-s + 42-s − 47-s − 48-s + 2·49-s − 53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\R}(s)^{2} \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(2563\)    =    \(11 \cdot 233\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 2563,\ (0, 0, 1:\ ),\ 1)$

Euler product

\[\begin{aligned}L(s,\rho) = \prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Particular Values

\[L(1/2,\rho) \approx 0.1773674555\] \[L(1,\rho) \approx 0.3789488055\]

Imaginary part of the first few zeros on the critical line

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