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Properties

Label 2-141120-1.1-c1-0-407
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·13-s − 6·17-s − 4·23-s + 25-s − 2·29-s − 8·31-s − 6·37-s − 6·41-s + 12·43-s + 12·47-s − 10·53-s + 8·59-s − 10·61-s + 2·65-s − 12·67-s + 8·71-s − 10·73-s − 16·79-s + 12·83-s + 6·85-s − 6·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.554·13-s − 1.45·17-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.986·37-s − 0.937·41-s + 1.82·43-s + 1.75·47-s − 1.37·53-s + 1.04·59-s − 1.28·61-s + 0.248·65-s − 1.46·67-s + 0.949·71-s − 1.17·73-s − 1.80·79-s + 1.31·83-s + 0.650·85-s − 0.635·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

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

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: $\chi_{141120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.71849192375269, −13.59060769838086, −12.74416093710340, −12.48298819696000, −12.05589018589424, −11.42187863514536, −11.03368111527733, −10.58030796940082, −10.14837038231726, −9.341316449737114, −9.109305042278337, −8.610790505616752, −8.011768355793310, −7.402793341479377, −7.153951728115012, −6.564681148904209, −5.873913385961777, −5.496818501168185, −4.772063891093019, −4.242921249752439, −3.892861676446197, −3.150865415880944, −2.481385876052743, −1.968476546531995, −1.264100128159062, 0, 0, 1.264100128159062, 1.968476546531995, 2.481385876052743, 3.150865415880944, 3.892861676446197, 4.242921249752439, 4.772063891093019, 5.496818501168185, 5.873913385961777, 6.564681148904209, 7.153951728115012, 7.402793341479377, 8.011768355793310, 8.610790505616752, 9.109305042278337, 9.341316449737114, 10.14837038231726, 10.58030796940082, 11.03368111527733, 11.42187863514536, 12.05589018589424, 12.48298819696000, 12.74416093710340, 13.59060769838086, 13.71849192375269

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