Properties

Label 2-20160-1.1-c1-0-60
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 + 2·11-s + 4·13-s + 2·17-s + 6·19-s + 4·23-s + 25-s − 10·29-s + 2·31-s − 35-s + 2·37-s + 10·41-s − 4·43-s + 8·47-s + 49-s + 4·53-s + 2·55-s + 4·59-s + 2·61-s + 4·65-s − 4·67-s − 6·71-s − 2·77-s − 4·79-s − 4·83-s + 2·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.603·11-s + 1.10·13-s + 0.485·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s − 1.85·29-s + 0.359·31-s − 0.169·35-s + 0.328·37-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.549·53-s + 0.269·55-s + 0.520·59-s + 0.256·61-s + 0.496·65-s − 0.488·67-s − 0.712·71-s − 0.227·77-s − 0.450·79-s − 0.439·83-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: $1$
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

\(L(1)\) \(\approx\) \(3.109665976\)
\(L(\frac12)\) \(\approx\) \(3.109665976\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−15.68457096839619, −15.12423914750925, −14.45695633965625, −14.07117613032547, −13.33873326443047, −13.11571336453219, −12.42864108839531, −11.67319371063765, −11.33738167280086, −10.68408150926627, −10.03964956543165, −9.415448934984064, −9.076034891548603, −8.476909541294977, −7.498306618765079, −7.270069069962307, −6.379448622917594, −5.806632057581882, −5.445534885643446, −4.496332576165618, −3.708049834204960, −3.260933508473273, −2.389148968067256, −1.398213661279340, −0.8103577327146951, 0.8103577327146951, 1.398213661279340, 2.389148968067256, 3.260933508473273, 3.708049834204960, 4.496332576165618, 5.445534885643446, 5.806632057581882, 6.379448622917594, 7.270069069962307, 7.498306618765079, 8.476909541294977, 9.076034891548603, 9.415448934984064, 10.03964956543165, 10.68408150926627, 11.33738167280086, 11.67319371063765, 12.42864108839531, 13.11571336453219, 13.33873326443047, 14.07117613032547, 14.45695633965625, 15.12423914750925, 15.68457096839619

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