Properties

Label 2-2160-1.1-c1-0-16
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 4·7-s + 2·11-s + 4·13-s − 17-s + 5·19-s + 5·23-s + 25-s − 8·29-s − 7·31-s + 4·35-s − 6·37-s − 6·41-s + 2·43-s + 8·47-s + 9·49-s − 9·53-s + 2·55-s + 4·59-s + 13·61-s + 4·65-s + 10·67-s − 6·71-s − 6·73-s + 8·77-s − 9·79-s − 17·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s + 0.603·11-s + 1.10·13-s − 0.242·17-s + 1.14·19-s + 1.04·23-s + 1/5·25-s − 1.48·29-s − 1.25·31-s + 0.676·35-s − 0.986·37-s − 0.937·41-s + 0.304·43-s + 1.16·47-s + 9/7·49-s − 1.23·53-s + 0.269·55-s + 0.520·59-s + 1.66·61-s + 0.496·65-s + 1.22·67-s − 0.712·71-s − 0.702·73-s + 0.911·77-s − 1.01·79-s − 1.86·83-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: $\chi_{2160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.615385370\)
\(L(\frac12)\) \(\approx\) \(2.615385370\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 17 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.925238629210729863020418287737, −8.478833179515949977669438635262, −7.46206699816106956360734624664, −6.88740520144549246250126905668, −5.64925836280054079608642247650, −5.28222711027526819045315544760, −4.20120998954151629074024789634, −3.33465948715580981375117818151, −1.90973494783978255742149684954, −1.22674050988694425207628828009, 1.22674050988694425207628828009, 1.90973494783978255742149684954, 3.33465948715580981375117818151, 4.20120998954151629074024789634, 5.28222711027526819045315544760, 5.64925836280054079608642247650, 6.88740520144549246250126905668, 7.46206699816106956360734624664, 8.478833179515949977669438635262, 8.925238629210729863020418287737

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