Properties

Label 2-20160-1.1-c1-0-9
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 4·11-s + 6·13-s − 6·17-s − 4·23-s + 25-s − 6·29-s − 35-s − 2·37-s − 2·41-s + 4·43-s + 4·47-s + 49-s − 6·53-s + 4·55-s − 12·59-s + 10·61-s − 6·65-s + 4·67-s + 8·71-s − 14·73-s − 4·77-s − 8·79-s + 12·83-s + 6·85-s + 6·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 1.20·11-s + 1.66·13-s − 1.45·17-s − 0.834·23-s + 1/5·25-s − 1.11·29-s − 0.169·35-s − 0.328·37-s − 0.312·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.824·53-s + 0.539·55-s − 1.56·59-s + 1.28·61-s − 0.744·65-s + 0.488·67-s + 0.949·71-s − 1.63·73-s − 0.455·77-s − 0.900·79-s + 1.31·83-s + 0.650·85-s + 0.635·89-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 20160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.313713566\)
\(L(\frac12)\) \(\approx\) \(1.313713566\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.68780209943023, −15.31174313361084, −14.63353997735427, −13.76692287751763, −13.62317912135271, −12.87846566658595, −12.55199822968841, −11.56973380629223, −11.24736093036685, −10.75968939531203, −10.31074367223582, −9.418951972591271, −8.773132911996432, −8.376308619251322, −7.774972183520364, −7.238145967255853, −6.405776837758270, −5.898595996380743, −5.208701523400590, −4.460622010167908, −3.905796505591965, −3.210442359273971, −2.295486137691901, −1.635907655983009, −0.4639208283800160, 0.4639208283800160, 1.635907655983009, 2.295486137691901, 3.210442359273971, 3.905796505591965, 4.460622010167908, 5.208701523400590, 5.898595996380743, 6.405776837758270, 7.238145967255853, 7.774972183520364, 8.376308619251322, 8.773132911996432, 9.418951972591271, 10.31074367223582, 10.75968939531203, 11.24736093036685, 11.56973380629223, 12.55199822968841, 12.87846566658595, 13.62317912135271, 13.76692287751763, 14.63353997735427, 15.31174313361084, 15.68780209943023

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