Properties

Label 2-2160-1.1-c1-0-30
Degree $2$
Conductor $2160$
Sign $-1$
Analytic cond. $17.2476$
Root an. cond. $4.15303$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 6·11-s − 13-s + 19-s − 6·23-s + 25-s + 6·29-s − 8·31-s + 35-s − 7·37-s − 6·41-s + 4·43-s − 12·47-s − 6·49-s − 6·53-s − 6·55-s + 11·61-s − 65-s + 7·67-s + 6·71-s + 11·73-s − 6·77-s + 79-s − 6·83-s − 12·89-s − 91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 1.80·11-s − 0.277·13-s + 0.229·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.169·35-s − 1.15·37-s − 0.937·41-s + 0.609·43-s − 1.75·47-s − 6/7·49-s − 0.824·53-s − 0.809·55-s + 1.40·61-s − 0.124·65-s + 0.855·67-s + 0.712·71-s + 1.28·73-s − 0.683·77-s + 0.112·79-s − 0.658·83-s − 1.27·89-s − 0.104·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-1$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ -1)\)

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 \( 1 - T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−8.453757101212738665043125480157, −8.054815367760551473988773648865, −7.20149128213989535337754984544, −6.30275477453639231316674547503, −5.26056570635904728739855325749, −4.99617429415613386423394911674, −3.65778919398785502430721530689, −2.61978163780340119595385627704, −1.76173777505214450808543092451, 0, 1.76173777505214450808543092451, 2.61978163780340119595385627704, 3.65778919398785502430721530689, 4.99617429415613386423394911674, 5.26056570635904728739855325749, 6.30275477453639231316674547503, 7.20149128213989535337754984544, 8.054815367760551473988773648865, 8.453757101212738665043125480157

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