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Properties

Label 2-87120-1.1-c1-0-133
Degree $2$
Conductor $87120$
Sign $-1$
Analytic cond. $695.656$
Root an. cond. $26.3753$
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 + 2·13-s + 6·17-s − 2·23-s + 25-s − 8·29-s − 4·31-s − 2·37-s − 4·41-s − 4·43-s + 2·47-s − 7·49-s + 10·53-s − 8·61-s + 2·65-s + 2·67-s + 8·71-s + 10·73-s − 4·79-s − 12·83-s + 6·85-s − 6·89-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.554·13-s + 1.45·17-s − 0.417·23-s + 1/5·25-s − 1.48·29-s − 0.718·31-s − 0.328·37-s − 0.624·41-s − 0.609·43-s + 0.291·47-s − 49-s + 1.37·53-s − 1.02·61-s + 0.248·65-s + 0.244·67-s + 0.949·71-s + 1.17·73-s − 0.450·79-s − 1.31·83-s + 0.650·85-s − 0.635·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87120\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(695.656\)
Root analytic conductor: \(26.3753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 87120,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 \)
good7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.09117176129310, −13.68786655098423, −13.21664723033038, −12.56684272694847, −12.37284646406733, −11.55069169663069, −11.28256181325796, −10.62433537429429, −10.12021699138294, −9.699799972700476, −9.242227862559271, −8.546612005301793, −8.213343600552288, −7.423706088815305, −7.201364983337997, −6.353984953026409, −5.909299002193268, −5.404089030804056, −4.980606850512468, −4.090714902330243, −3.538155144901061, −3.153564952450545, −2.189719170337280, −1.681124480138527, −0.9921975660953208, 0, 0.9921975660953208, 1.681124480138527, 2.189719170337280, 3.153564952450545, 3.538155144901061, 4.090714902330243, 4.980606850512468, 5.404089030804056, 5.909299002193268, 6.353984953026409, 7.201364983337997, 7.423706088815305, 8.213343600552288, 8.546612005301793, 9.242227862559271, 9.699799972700476, 10.12021699138294, 10.62433537429429, 11.28256181325796, 11.55069169663069, 12.37284646406733, 12.56684272694847, 13.21664723033038, 13.68786655098423, 14.09117176129310

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