Properties

Degree 3
Conductor $ 2^{2} \cdot 373 $
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 − 5-s − 7-s + 8-s − 10-s − 11-s − 14-s + 16-s − 19-s − 20-s − 22-s + 23-s + 27-s − 28-s − 29-s + 32-s + 35-s − 38-s − 40-s + 43-s − 44-s + 46-s − 47-s + 2·49-s − 53-s + 54-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1492 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(1492\)    =    \(2^{2} \cdot 373\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 1492,\ (0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 0.9390987926\] \[L(1,\rho) \approx 1.267498875\]

Imaginary part of the first few zeros on the critical line

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