Properties

Label 2-121680-1.1-c1-0-48
Degree $2$
Conductor $121680$
Sign $1$
Analytic cond. $971.619$
Root an. cond. $31.1708$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 2·7-s − 4·11-s + 4·17-s + 6·19-s + 25-s + 4·29-s + 6·31-s − 2·35-s + 2·37-s + 10·41-s − 8·43-s − 3·49-s + 4·53-s + 4·55-s − 4·59-s + 2·61-s + 6·67-s + 8·71-s − 10·73-s − 8·77-s + 4·79-s + 12·83-s − 4·85-s − 2·89-s − 6·95-s − 2·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s − 1.20·11-s + 0.970·17-s + 1.37·19-s + 1/5·25-s + 0.742·29-s + 1.07·31-s − 0.338·35-s + 0.328·37-s + 1.56·41-s − 1.21·43-s − 3/7·49-s + 0.549·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.733·67-s + 0.949·71-s − 1.17·73-s − 0.911·77-s + 0.450·79-s + 1.31·83-s − 0.433·85-s − 0.211·89-s − 0.615·95-s − 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121680\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(971.619\)
Root analytic conductor: \(31.1708\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{121680} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 121680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.956041964\)
\(L(\frac12)\) \(\approx\) \(2.956041964\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.57365377188804, −13.14027719506526, −12.42721431297094, −12.14160076085118, −11.62384980445826, −11.15773716661699, −10.72571490776468, −10.06208025042174, −9.821573332311572, −9.184454252968022, −8.385443713612966, −8.106086815619284, −7.720710168280708, −7.269359281503079, −6.638309609286651, −5.871723910667447, −5.462088744141479, −4.842242031098704, −4.599068501217231, −3.727229699713981, −3.121709619103650, −2.705824835871637, −1.935131648453142, −1.080005500294733, −0.6093685390143791, 0.6093685390143791, 1.080005500294733, 1.935131648453142, 2.705824835871637, 3.121709619103650, 3.727229699713981, 4.599068501217231, 4.842242031098704, 5.462088744141479, 5.871723910667447, 6.638309609286651, 7.269359281503079, 7.720710168280708, 8.106086815619284, 8.385443713612966, 9.184454252968022, 9.821573332311572, 10.06208025042174, 10.72571490776468, 11.15773716661699, 11.62384980445826, 12.14160076085118, 12.42721431297094, 13.14027719506526, 13.57365377188804

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