Properties

Degree 3
Conductor $ 2^{2} \cdot 5 \cdot 443 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  − 3-s + 4-s + 2·9-s − 12-s + 16-s − 25-s − 2·27-s − 29-s + 2·36-s − 41-s + 43-s + 47-s − 48-s − 53-s + 59-s + 64-s + 67-s + 71-s − 73-s + 75-s − 79-s + 3·81-s + 87-s + 97-s − 100-s − 101-s − 103-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8860 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(8860\)    =    \(2^{2} \cdot 5 \cdot 443\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 8860,\ (0, 1, 1:\ ),\ 1)$

Euler product

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

Particular Values

\[L(1/2,\rho) \approx 1.017712097\] \[L(1,\rho) \approx 1.023798824\]

Imaginary part of the first few zeros on the critical line

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