Properties

Label 2-87120-1.1-c1-0-122
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 + 7-s + 6·13-s + 3·17-s − 5·19-s − 2·23-s + 25-s − 5·29-s − 5·31-s − 35-s − 37-s − 2·41-s + 12·43-s − 2·47-s − 6·49-s + 13·53-s + 2·59-s − 61-s − 6·65-s − 16·67-s + 15·71-s − 10·73-s + 2·79-s + 14·83-s − 3·85-s − 9·89-s + 6·91-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 1.66·13-s + 0.727·17-s − 1.14·19-s − 0.417·23-s + 1/5·25-s − 0.928·29-s − 0.898·31-s − 0.169·35-s − 0.164·37-s − 0.312·41-s + 1.82·43-s − 0.291·47-s − 6/7·49-s + 1.78·53-s + 0.260·59-s − 0.128·61-s − 0.744·65-s − 1.95·67-s + 1.78·71-s − 1.17·73-s + 0.225·79-s + 1.53·83-s − 0.325·85-s − 0.953·89-s + 0.628·91-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: $\chi_{87120} (1, \cdot )$
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 \( 1 - T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
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

−14.06705335051340, −13.75549852352639, −13.00316144732493, −12.81605568769344, −12.14827440928610, −11.62342814014048, −11.15513453767838, −10.71207796138373, −10.39536119440419, −9.604354741592687, −8.979766345752114, −8.672545263630735, −8.069511104519693, −7.684557779120058, −7.069808259579961, −6.451270853823614, −5.844657866525187, −5.542926789974398, −4.728360610364972, −3.966936195706456, −3.839739066400467, −3.103613559812108, −2.249688725792683, −1.608664874873281, −0.9441137457846837, 0, 0.9441137457846837, 1.608664874873281, 2.249688725792683, 3.103613559812108, 3.839739066400467, 3.966936195706456, 4.728360610364972, 5.542926789974398, 5.844657866525187, 6.451270853823614, 7.069808259579961, 7.684557779120058, 8.069511104519693, 8.672545263630735, 8.979766345752114, 9.604354741592687, 10.39536119440419, 10.71207796138373, 11.15513453767838, 11.62342814014048, 12.14827440928610, 12.81605568769344, 13.00316144732493, 13.75549852352639, 14.06705335051340

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