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

\(L(1)\) \(\approx\) \(1.369577023\)
\(L(\frac12)\) \(\approx\) \(1.369577023\)
\(L(\frac{3}{2})\) 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

−15.60545398088059, −15.32123261791534, −14.50803840726278, −13.85690387205636, −13.52339133150009, −12.94679712982385, −12.53980657305384, −11.72933617984587, −11.31825143750589, −10.42827002110341, −10.10093471620563, −9.822183080769202, −8.693039622691323, −8.409033168442830, −7.907678993089947, −6.993602102615927, −6.303420857356491, −6.099069599463242, −5.149117761604585, −4.623895153634463, −3.865591904871275, −3.016478067901984, −2.388351666833144, −1.680629997045177, −0.4634765892038448, 0.4634765892038448, 1.680629997045177, 2.388351666833144, 3.016478067901984, 3.865591904871275, 4.623895153634463, 5.149117761604585, 6.099069599463242, 6.303420857356491, 6.993602102615927, 7.907678993089947, 8.409033168442830, 8.693039622691323, 9.822183080769202, 10.10093471620563, 10.42827002110341, 11.31825143750589, 11.72933617984587, 12.53980657305384, 12.94679712982385, 13.52339133150009, 13.85690387205636, 14.50803840726278, 15.32123261791534, 15.60545398088059

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