Properties

Degree 3
Conductor $ 3^{6} \cdot 11^{2} $
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  − 0.618·2-s + 0.999·4-s − 7-s + 11-s + 1.61·13-s + 0.618·14-s + 1.61·17-s − 0.618·22-s + 1.61·23-s − 0.999·26-s − 0.999·28-s + 1.00·32-s − 0.999·34-s − 0.618·37-s − 0.618·43-s + 0.999·44-s − 0.999·46-s + 1.61·47-s + 2·49-s + 1.61·52-s − 0.618·53-s − 0.618·59-s − 61-s − 0.618·64-s + 1.61·68-s − 71-s − 0.618·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88209 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(88209\)    =    \(3^{6} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((3,\ 88209,\ (0, 1, 1:\ ),\ 1)\)

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 1.626308326\] \[L(1,\rho) \approx 1.113590000\]

Imaginary part of the first few zeros on the critical line

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