Properties

Label 2-1080-120.29-c0-0-3
Degree $2$
Conductor $1080$
Sign $1$
Analytic cond. $0.538990$
Root an. cond. $0.734159$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s − 13-s + 16-s + 17-s − 20-s − 22-s + 23-s + 25-s + 26-s + 29-s − 31-s − 32-s − 34-s + 2·37-s + 40-s − 43-s + 44-s − 46-s + 47-s + 49-s − 50-s − 52-s + ⋯
L(s)  = 1  − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s − 13-s + 16-s + 17-s − 20-s − 22-s + 23-s + 25-s + 26-s + 29-s − 31-s − 32-s − 34-s + 2·37-s + 40-s − 43-s + 44-s − 46-s + 47-s + 49-s − 50-s − 52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.538990\)
Root analytic conductor: \(0.734159\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1080} (269, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5953219492\)
\(L(\frac12)\) \(\approx\) \(0.5953219492\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
good7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 - T + T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 - T + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 + T )^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.932533316440766670597905423549, −9.238439906726089955707863892086, −8.462560489209064748218494141714, −7.57761370109169708088616904981, −7.10608373475732770455737214864, −6.11865384691372893935298573913, −4.87298187136596254241737222535, −3.68451450330355070794671732683, −2.68464939889574145993621851990, −1.06151035905282882400200545743, 1.06151035905282882400200545743, 2.68464939889574145993621851990, 3.68451450330355070794671732683, 4.87298187136596254241737222535, 6.11865384691372893935298573913, 7.10608373475732770455737214864, 7.57761370109169708088616904981, 8.462560489209064748218494141714, 9.238439906726089955707863892086, 9.932533316440766670597905423549

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