Properties

Degree 3
Conductor 229
Sign $1$
Self-dual yes
Motivic weight 0

Related objects

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Normalization:  

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Dirichlet series

$L(s,\rho)$  = 1  − 2-s − 7-s − 13-s + 14-s + 16-s + 23-s + 26-s + 27-s + 29-s − 31-s − 32-s − 37-s − 41-s − 46-s + 47-s − 53-s − 54-s − 58-s − 59-s + 62-s − 67-s + 73-s + 74-s + 79-s + 82-s + 89-s + 91-s + ⋯

Functional equation

\[\begin{align} \Lambda(s)=\mathstrut & 229 ^{s/2} \Gamma_{\R}(s) \Gamma_{\R}(s+1) ^{2} \cdot L(s,\rho)\cr =\mathstrut & \Lambda(1-s) \end{align} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(229\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 229,\ (0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.2157489919\] \[L(1,\rho) \approx 0.4400768328\]

Imaginary part of the first few zeros on the critical line

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