Properties

Degree 3
Conductor $ 3^{2} \cdot 61^{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  − 3-s + 8-s + 9-s − 11-s − 23-s − 24-s − 27-s + 33-s − 37-s − 41-s − 53-s + 61-s + 64-s + 69-s + 72-s + 81-s − 88-s − 89-s − 99-s + 111-s − 113-s + 2·121-s + 123-s + 125-s − 131-s − 149-s + 159-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 33489 ^{s/2} \, \Gamma_{\R}(s)^{3} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(33489\)    =    \(3^{2} \cdot 61^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 33489,\ (0, 0, 0:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.3461267618\] \[L(1,\rho) \approx 0.6960086871\]

Imaginary part of the first few zeros on the critical line

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