Properties

Label 2-1440-1.1-c1-0-13
Degree $2$
Conductor $1440$
Sign $-1$
Analytic cond. $11.4984$
Root an. cond. $3.39093$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2.82·7-s + 5.65·11-s − 2·13-s − 2·17-s − 2.82·23-s + 25-s − 6·29-s + 5.65·31-s + 2.82·35-s − 10·37-s − 2·41-s − 8.48·43-s + 2.82·47-s + 1.00·49-s − 6·53-s − 5.65·55-s − 11.3·59-s − 2·61-s + 2·65-s − 2.82·67-s − 5.65·71-s − 6·73-s − 16.0·77-s + 11.3·79-s − 2.82·83-s + 2·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.06·7-s + 1.70·11-s − 0.554·13-s − 0.485·17-s − 0.589·23-s + 0.200·25-s − 1.11·29-s + 1.01·31-s + 0.478·35-s − 1.64·37-s − 0.312·41-s − 1.29·43-s + 0.412·47-s + 0.142·49-s − 0.824·53-s − 0.762·55-s − 1.47·59-s − 0.256·61-s + 0.248·65-s − 0.345·67-s − 0.671·71-s − 0.702·73-s − 1.82·77-s + 1.27·79-s − 0.310·83-s + 0.216·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-1$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
good7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 2.82T + 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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

−9.206883926168802619577668862186, −8.459899785669851719256164874973, −7.35586860808036520391291120827, −6.66330364534288339791213305386, −6.07074588456278407762999436348, −4.77931011777860238276576962090, −3.86485250752345530176479745830, −3.14880370233397003691309117407, −1.69086024587047792801753188621, 0, 1.69086024587047792801753188621, 3.14880370233397003691309117407, 3.86485250752345530176479745830, 4.77931011777860238276576962090, 6.07074588456278407762999436348, 6.66330364534288339791213305386, 7.35586860808036520391291120827, 8.459899785669851719256164874973, 9.206883926168802619577668862186

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