Properties

Label 2-87120-1.1-c1-0-114
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 + 2·7-s + 4·13-s − 6·17-s − 4·19-s + 25-s − 6·29-s + 4·31-s − 2·35-s + 8·37-s + 8·43-s − 3·49-s + 6·53-s + 6·59-s − 2·61-s − 4·65-s + 4·67-s − 12·71-s + 10·73-s − 4·79-s − 12·83-s + 6·85-s − 12·89-s + 8·91-s + 4·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s + 1.10·13-s − 1.45·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.338·35-s + 1.31·37-s + 1.21·43-s − 3/7·49-s + 0.824·53-s + 0.781·59-s − 0.256·61-s − 0.496·65-s + 0.488·67-s − 1.42·71-s + 1.17·73-s − 0.450·79-s − 1.31·83-s + 0.650·85-s − 1.27·89-s + 0.838·91-s + 0.410·95-s + 0.203·97-s + 0.0995·101-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.18082423315262, −13.56768709497715, −13.15672249456885, −12.79955857730104, −12.14295660734560, −11.47264384624217, −11.14401470844172, −10.94476749316538, −10.29964485923242, −9.589365073895468, −9.005655724075330, −8.587047863668166, −8.164808247349429, −7.681032131386023, −6.967611377294452, −6.551226239620947, −5.909827356237361, −5.447312546141591, −4.569501964043816, −4.220543043316389, −3.864920419650690, −2.902564913934536, −2.322566044100965, −1.651600502559343, −0.9088148584801605, 0, 0.9088148584801605, 1.651600502559343, 2.322566044100965, 2.902564913934536, 3.864920419650690, 4.220543043316389, 4.569501964043816, 5.447312546141591, 5.909827356237361, 6.551226239620947, 6.967611377294452, 7.681032131386023, 8.164808247349429, 8.587047863668166, 9.005655724075330, 9.589365073895468, 10.29964485923242, 10.94476749316538, 11.14401470844172, 11.47264384624217, 12.14295660734560, 12.79955857730104, 13.15672249456885, 13.56768709497715, 14.18082423315262

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