Properties

Label 2-87120-1.1-c1-0-113
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 + 3·7-s − 2·13-s − 6·17-s + 5·19-s + 4·23-s + 25-s − 6·29-s + 5·31-s − 3·35-s − 11·37-s − 4·41-s + 4·43-s − 6·47-s + 2·49-s + 8·53-s − 2·59-s + 61-s + 2·65-s + 7·67-s − 10·71-s − 11·73-s − 79-s + 10·83-s + 6·85-s + 14·89-s − 6·91-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.13·7-s − 0.554·13-s − 1.45·17-s + 1.14·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s + 0.898·31-s − 0.507·35-s − 1.80·37-s − 0.624·41-s + 0.609·43-s − 0.875·47-s + 2/7·49-s + 1.09·53-s − 0.260·59-s + 0.128·61-s + 0.248·65-s + 0.855·67-s − 1.18·71-s − 1.28·73-s − 0.112·79-s + 1.09·83-s + 0.650·85-s + 1.48·89-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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−14.12636471438472, −13.66688221305267, −13.24841991249083, −12.65615329705550, −12.02147857844819, −11.58343674391326, −11.36638717024578, −10.69523626818537, −10.33793846438601, −9.575753969695416, −9.076264889926215, −8.584716039543947, −8.157873877694013, −7.488102021694620, −7.116923889991874, −6.694734753719993, −5.792279793655095, −5.263481197681117, −4.725305106614791, −4.430255922404482, −3.564291933861182, −3.063603074851116, −2.203500159281768, −1.727702436633653, −0.8956699450148909, 0, 0.8956699450148909, 1.727702436633653, 2.203500159281768, 3.063603074851116, 3.564291933861182, 4.430255922404482, 4.725305106614791, 5.263481197681117, 5.792279793655095, 6.694734753719993, 7.116923889991874, 7.488102021694620, 8.157873877694013, 8.584716039543947, 9.076264889926215, 9.575753969695416, 10.33793846438601, 10.69523626818537, 11.36638717024578, 11.58343674391326, 12.02147857844819, 12.65615329705550, 13.24841991249083, 13.66688221305267, 14.12636471438472

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