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 + 6·13-s − 2·17-s + 25-s + 6·29-s + 8·31-s + 35-s + 10·37-s − 2·41-s − 4·43-s − 8·47-s + 49-s − 2·53-s − 4·55-s − 8·59-s + 14·61-s − 6·65-s + 12·67-s + 16·71-s + 2·73-s − 4·77-s − 8·79-s + 8·83-s + 2·85-s − 10·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 1.20·11-s + 1.66·13-s − 0.485·17-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.169·35-s + 1.64·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s − 0.539·55-s − 1.04·59-s + 1.79·61-s − 0.744·65-s + 1.46·67-s + 1.89·71-s + 0.234·73-s − 0.455·77-s − 0.900·79-s + 0.878·83-s + 0.216·85-s − 1.05·89-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

\(L(1)\) \(\approx\) \(2.559505658\)
\(L(\frac12)\) \(\approx\) \(2.559505658\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.63849773876631, −15.31626516797089, −14.43240008067825, −14.09870222465945, −13.43180387543223, −12.98689996589128, −12.36439548108731, −11.60635213091742, −11.40534296912298, −10.81553470467139, −9.981567531318383, −9.582689701167671, −8.797071991058377, −8.373819634415930, −7.930561923914175, −6.877945442645915, −6.389003823542150, −6.232040771882118, −5.124690347904054, −4.404317255567873, −3.830536110816996, −3.298957216991220, −2.433790063196622, −1.336676027019226, −0.7465416445234451, 0.7465416445234451, 1.336676027019226, 2.433790063196622, 3.298957216991220, 3.830536110816996, 4.404317255567873, 5.124690347904054, 6.232040771882118, 6.389003823542150, 6.877945442645915, 7.930561923914175, 8.373819634415930, 8.797071991058377, 9.582689701167671, 9.981567531318383, 10.81553470467139, 11.40534296912298, 11.60635213091742, 12.36439548108731, 12.98689996589128, 13.43180387543223, 14.09870222465945, 14.43240008067825, 15.31626516797089, 15.63849773876631

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