↑

Properties

Label 2-55440-1.1-c1-0-105
Degree $2$
Conductor $55440$
Sign $-1$
Analytic cond. $442.690$
Root an. cond. $21.0402$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 11-s + 2·13-s − 2·17-s − 4·19-s − 4·23-s + 25-s + 6·29-s + 35-s + 2·37-s − 6·41-s − 12·43-s + 8·47-s + 49-s + 6·53-s − 55-s + 8·59-s − 14·61-s + 2·65-s − 12·67-s + 8·71-s + 10·73-s − 77-s − 4·79-s + 16·83-s − 2·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.169·35-s + 0.328·37-s − 0.937·41-s − 1.82·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s − 0.134·55-s + 1.04·59-s − 1.79·61-s + 0.248·65-s − 1.46·67-s + 0.949·71-s + 1.17·73-s − 0.113·77-s − 0.450·79-s + 1.75·83-s − 0.216·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55440\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(442.690\)
Root analytic conductor: \(21.0402\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 55440,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.70330411749105, −14.01224517787495, −13.58899311904482, −13.33808035892092, −12.62945076894395, −12.03539445252062, −11.73470219067402, −10.89149011758619, −10.61342094387499, −10.12369185985589, −9.529833770964089, −8.868595066939766, −8.360060996781072, −8.104055436607659, −7.239370519316187, −6.677179012571767, −6.195727828120897, −5.664057770599362, −4.942280962058602, −4.482030925856679, −3.814044826116761, −3.106747170466535, −2.306576883969274, −1.852245065681909, −1.002594207558674, 0, 1.002594207558674, 1.852245065681909, 2.306576883969274, 3.106747170466535, 3.814044826116761, 4.482030925856679, 4.942280962058602, 5.664057770599362, 6.195727828120897, 6.677179012571767, 7.239370519316187, 8.104055436607659, 8.360060996781072, 8.868595066939766, 9.529833770964089, 10.12369185985589, 10.61342094387499, 10.89149011758619, 11.73470219067402, 12.03539445252062, 12.62945076894395, 13.33808035892092, 13.58899311904482, 14.01224517787495, 14.70330411749105

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