Properties

Label 2-20160-1.1-c1-0-81
Degree $2$
Conductor $20160$
Sign $-1$
Analytic cond. $160.978$
Root an. cond. $12.6877$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s + 3·11-s − 5·13-s − 3·17-s + 2·19-s − 6·23-s + 25-s + 3·29-s + 4·31-s + 35-s − 2·37-s + 12·41-s − 10·43-s + 9·47-s + 49-s + 12·53-s − 3·55-s − 8·61-s + 5·65-s − 4·67-s + 2·73-s − 3·77-s + 79-s − 12·83-s + 3·85-s + 12·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 0.904·11-s − 1.38·13-s − 0.727·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s + 0.557·29-s + 0.718·31-s + 0.169·35-s − 0.328·37-s + 1.87·41-s − 1.52·43-s + 1.31·47-s + 1/7·49-s + 1.64·53-s − 0.404·55-s − 1.02·61-s + 0.620·65-s − 0.488·67-s + 0.234·73-s − 0.341·77-s + 0.112·79-s − 1.31·83-s + 0.325·85-s + 1.27·89-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: $-1$
Analytic conductor: \(160.978\)
Root analytic conductor: \(12.6877\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{20160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 20160,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
good11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−15.97715274404588, −15.36952797494398, −14.83945845380390, −14.34237364891917, −13.75495285587948, −13.30199006424507, −12.32734760676798, −12.17402968395493, −11.71587711143152, −10.99785736890617, −10.27329458834345, −9.841051868164652, −9.235255068576990, −8.691060868681023, −7.994573831200012, −7.337819282688609, −6.908930520909077, −6.210998942700925, −5.608636476454559, −4.667549310881372, −4.284778172470667, −3.559675132912979, −2.709076890803274, −2.097821018748656, −0.9697620591159515, 0, 0.9697620591159515, 2.097821018748656, 2.709076890803274, 3.559675132912979, 4.284778172470667, 4.667549310881372, 5.608636476454559, 6.210998942700925, 6.908930520909077, 7.337819282688609, 7.994573831200012, 8.691060868681023, 9.235255068576990, 9.841051868164652, 10.27329458834345, 10.99785736890617, 11.71587711143152, 12.17402968395493, 12.32734760676798, 13.30199006424507, 13.75495285587948, 14.34237364891917, 14.83945845380390, 15.36952797494398, 15.97715274404588

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