Properties

Degree 3
Conductor $ 2^{2} \cdot 3^{4} \cdot 11^{2} $
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 + 4-s + 1.61·5-s − 8-s − 1.61·10-s + 11-s + 1.61·13-s + 16-s − 0.618·17-s + 1.61·20-s − 22-s − 0.618·23-s + 25-s − 1.61·26-s − 32-s + 0.618·34-s − 37-s − 1.61·40-s − 43-s + 44-s + 0.618·46-s + 1.61·47-s − 50-s + 1.61·52-s + 1.61·55-s + 1.61·61-s + 64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 39204 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(39204\)    =    \(2^{2} \cdot 3^{4} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 39204,\ (0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 1.310066331\] \[L(1,\rho) \approx 1.018078649\]

Imaginary part of the first few zeros on the critical line

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