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Properties

Label 2-262080-1.1-c1-0-103
Degree $2$
Conductor $262080$
Sign $-1$
Analytic cond. $2092.71$
Root an. cond. $45.7462$
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 − 4·11-s + 13-s − 6·17-s − 4·23-s + 25-s − 2·29-s + 4·31-s + 35-s − 2·37-s + 2·41-s − 8·43-s + 8·47-s + 49-s − 2·53-s + 4·55-s − 8·59-s − 6·61-s − 65-s + 12·67-s + 6·73-s + 4·77-s − 8·79-s − 4·83-s + 6·85-s + 10·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.20·11-s + 0.277·13-s − 1.45·17-s − 0.834·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.169·35-s − 0.328·37-s + 0.312·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s − 0.274·53-s + 0.539·55-s − 1.04·59-s − 0.768·61-s − 0.124·65-s + 1.46·67-s + 0.702·73-s + 0.455·77-s − 0.900·79-s − 0.439·83-s + 0.650·85-s + 1.05·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(262080\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(2092.71\)
Root analytic conductor: \(45.7462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 262080,\ (\ :1/2),\ -1)\)

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

−13.09211693059272, −12.60555466135988, −12.15716336508083, −11.63084912095692, −11.17838189305965, −10.71408776836449, −10.37487429267449, −9.837619009794234, −9.342569546160819, −8.738875105174266, −8.433060130088392, −7.869390458059846, −7.471901550156572, −6.948014055108060, −6.301875152661956, −6.103544011762484, −5.282551393767532, −4.911876768732248, −4.325948434957667, −3.839967992369155, −3.267805195321255, −2.623731422201484, −2.224445434492358, −1.517493743665043, −0.5511021146114208, 0, 0.5511021146114208, 1.517493743665043, 2.224445434492358, 2.623731422201484, 3.267805195321255, 3.839967992369155, 4.325948434957667, 4.911876768732248, 5.282551393767532, 6.103544011762484, 6.301875152661956, 6.948014055108060, 7.471901550156572, 7.869390458059846, 8.433060130088392, 8.738875105174266, 9.342569546160819, 9.837619009794234, 10.37487429267449, 10.71408776836449, 11.17838189305965, 11.63084912095692, 12.15716336508083, 12.60555466135988, 13.09211693059272

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