Properties

Label 2-141120-1.1-c1-0-168
Degree $2$
Conductor $141120$
Sign $-1$
Analytic cond. $1126.84$
Root an. cond. $33.5685$
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 − 4·11-s − 2·13-s + 2·17-s − 4·19-s + 25-s − 10·29-s − 6·37-s − 6·41-s + 4·43-s − 8·47-s + 6·53-s + 4·55-s + 4·59-s − 10·61-s + 2·65-s − 4·67-s + 16·71-s + 14·73-s + 8·79-s + 4·83-s − 2·85-s + 10·89-s + 4·95-s − 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 1/5·25-s − 1.85·29-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 0.824·53-s + 0.539·55-s + 0.520·59-s − 1.28·61-s + 0.248·65-s − 0.488·67-s + 1.89·71-s + 1.63·73-s + 0.900·79-s + 0.439·83-s − 0.216·85-s + 1.05·89-s + 0.410·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(141120\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(1126.84\)
Root analytic conductor: \(33.5685\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 141120,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.61091909393483, −13.05550429397015, −12.74861672035111, −12.18278446963718, −11.86254136605327, −11.13147187601282, −10.70122486437883, −10.49459472178761, −9.647653692290048, −9.474147529062951, −8.693860722105408, −8.180215222420981, −7.849179847756566, −7.319464354196773, −6.843974110176879, −6.246934699053411, −5.530111536886670, −5.160371567982040, −4.721346099435654, −3.893094152749608, −3.536767359958356, −2.856383889212227, −2.165787910322566, −1.746724513270904, −0.6161420993904838, 0, 0.6161420993904838, 1.746724513270904, 2.165787910322566, 2.856383889212227, 3.536767359958356, 3.893094152749608, 4.721346099435654, 5.160371567982040, 5.530111536886670, 6.246934699053411, 6.843974110176879, 7.319464354196773, 7.849179847756566, 8.180215222420981, 8.693860722105408, 9.474147529062951, 9.647653692290048, 10.49459472178761, 10.70122486437883, 11.13147187601282, 11.86254136605327, 12.18278446963718, 12.74861672035111, 13.05550429397015, 13.61091909393483

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