Properties

Degree 3
Conductor $ 3^{6} \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  − 0.618·5-s + 8-s + 11-s − 0.618·13-s − 17-s + 1.61·23-s + 0.999·25-s − 29-s + 1.61·31-s − 0.618·37-s − 0.618·40-s + 1.61·41-s − 43-s − 0.618·55-s + 1.61·59-s − 61-s + 64-s + 0.381·65-s − 67-s − 0.618·71-s + 0.618·85-s + 88-s − 97-s − 0.618·101-s + 1.61·103-s − 0.618·104-s − 0.618·107-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

\[L(1/2,\rho) \approx 1.524263724\] \[L(1,\rho) \approx 1.113794457\]

Imaginary part of the first few zeros on the critical line

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