Properties

Degree $2$
Conductor $20160$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s + 8·23-s + 25-s + 6·29-s − 8·31-s + 35-s + 2·37-s − 2·41-s + 12·43-s + 8·47-s + 49-s + 6·53-s + 4·55-s + 4·59-s + 2·61-s + 2·65-s − 12·67-s − 8·71-s − 14·73-s + 4·77-s + 12·83-s − 2·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.169·35-s + 0.328·37-s − 0.312·41-s + 1.82·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s + 0.248·65-s − 1.46·67-s − 0.949·71-s − 1.63·73-s + 0.455·77-s + 1.31·83-s − 0.216·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20160\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{20160} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 20160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.568887725\)
\(L(\frac12)\) \(\approx\) \(3.568887725\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
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 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−15.66801461668974, −14.96772533140457, −14.57230995141674, −14.04335366487783, −13.50501445515472, −13.02262056972428, −12.32876373094897, −11.76038455534062, −11.25309600040401, −10.71903309617147, −10.17109091807806, −9.283643285582804, −8.952868214964111, −8.649298499694827, −7.446664462844097, −7.267880856873553, −6.435397181953344, −5.892086777844393, −5.246465035980816, −4.523209275913253, −3.896850243745391, −3.123152696208963, −2.366004526010983, −1.381442210075654, −0.8917103692719907, 0.8917103692719907, 1.381442210075654, 2.366004526010983, 3.123152696208963, 3.896850243745391, 4.523209275913253, 5.246465035980816, 5.892086777844393, 6.435397181953344, 7.267880856873553, 7.446664462844097, 8.649298499694827, 8.952868214964111, 9.283643285582804, 10.17109091807806, 10.71903309617147, 11.25309600040401, 11.76038455534062, 12.32876373094897, 13.02262056972428, 13.50501445515472, 14.04335366487783, 14.57230995141674, 14.96772533140457, 15.66801461668974

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