Properties

Label 2-2160-1.1-c1-0-21
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 − 2·7-s + 4·11-s − 2·13-s − 5·17-s + 5·19-s + 23-s + 25-s + 2·29-s − 7·31-s + 2·35-s − 6·37-s − 4·43-s + 4·47-s − 3·49-s − 9·53-s − 4·55-s + 14·59-s − 11·61-s + 2·65-s − 14·67-s − 12·73-s − 8·77-s + 3·79-s − 83-s + 5·85-s + 4·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s + 1.20·11-s − 0.554·13-s − 1.21·17-s + 1.14·19-s + 0.208·23-s + 1/5·25-s + 0.371·29-s − 1.25·31-s + 0.338·35-s − 0.986·37-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 1.23·53-s − 0.539·55-s + 1.82·59-s − 1.40·61-s + 0.248·65-s − 1.71·67-s − 1.40·73-s − 0.911·77-s + 0.337·79-s − 0.109·83-s + 0.542·85-s + 0.419·91-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: $\chi_{2160} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−8.987750842725442006551373585122, −7.83561238793477898112972070751, −6.99079300610682494080704471036, −6.56644807720865578285045226703, −5.50896552798411729795788051275, −4.54531903488051111371354188613, −3.70443672612966276007439183421, −2.89711829476831056635952255938, −1.54336384157991496196857038231, 0, 1.54336384157991496196857038231, 2.89711829476831056635952255938, 3.70443672612966276007439183421, 4.54531903488051111371354188613, 5.50896552798411729795788051275, 6.56644807720865578285045226703, 6.99079300610682494080704471036, 7.83561238793477898112972070751, 8.987750842725442006551373585122

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