Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·7-s + 9-s + 4·13-s + 6·17-s − 4·19-s − 2·21-s − 6·23-s − 5·25-s − 27-s − 6·29-s − 8·31-s − 10·37-s − 4·39-s − 6·41-s + 8·43-s + 6·47-s − 3·49-s − 6·51-s + 4·57-s − 8·61-s + 2·63-s + 4·67-s + 6·69-s − 6·71-s − 2·73-s + 5·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.10·13-s + 1.45·17-s − 0.917·19-s − 0.436·21-s − 1.25·23-s − 25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 1.64·37-s − 0.640·39-s − 0.937·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s − 0.840·51-s + 0.529·57-s − 1.02·61-s + 0.251·63-s + 0.488·67-s + 0.722·69-s − 0.712·71-s − 0.234·73-s + 0.577·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5808\)    =    \(2^{4} \cdot 3 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{5808} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 5808,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
11 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.89295470468755, −17.07969080387354, −16.74745696500886, −15.99941892361571, −15.50977468060600, −14.78977925255675, −14.14625790053543, −13.69031282110193, −12.83324181615387, −12.27172585809529, −11.70524748942774, −11.03525493716154, −10.56954362475105, −9.909588703892126, −9.077314797301423, −8.371363580509343, −7.728182326402871, −7.150011512280727, −6.014358818705124, −5.778596189092536, −4.975893845324225, −3.941152562973433, −3.563421257586685, −2.022922539422580, −1.426705292064923, 0, 1.426705292064923, 2.022922539422580, 3.563421257586685, 3.941152562973433, 4.975893845324225, 5.778596189092536, 6.014358818705124, 7.150011512280727, 7.728182326402871, 8.371363580509343, 9.077314797301423, 9.909588703892126, 10.56954362475105, 11.03525493716154, 11.70524748942774, 12.27172585809529, 12.83324181615387, 13.69031282110193, 14.14625790053543, 14.78977925255675, 15.50977468060600, 15.99941892361571, 16.74745696500886, 17.07969080387354, 17.89295470468755

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