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Properties

Label 2-262080-1.1-c1-0-317
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 6·11-s − 13-s − 6·17-s + 2·19-s + 4·23-s + 25-s − 8·29-s + 8·31-s + 35-s − 2·37-s − 2·41-s − 4·43-s − 4·47-s + 49-s − 6·53-s + 6·55-s + 8·61-s + 65-s + 2·67-s − 12·71-s + 2·73-s + 6·77-s − 4·79-s − 10·83-s + 6·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.80·11-s − 0.277·13-s − 1.45·17-s + 0.458·19-s + 0.834·23-s + 1/5·25-s − 1.48·29-s + 1.43·31-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 + 0.809·55-s + 1.02·61-s + 0.124·65-s + 0.244·67-s − 1.42·71-s + 0.234·73-s + 0.683·77-s − 0.450·79-s − 1.09·83-s + 0.650·85-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: \(2\)
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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
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 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.17950630000705, −12.93650780613424, −12.48131283682378, −11.80248412184591, −11.33981216696942, −11.06868988388294, −10.51815495008167, −9.985208662988923, −9.746110466030916, −8.989522494530692, −8.580704235254225, −8.198833490629829, −7.506247171927843, −7.312875713787034, −6.729900099265949, −6.189012896429537, −5.604538326791249, −4.981523469925285, −4.792540375346400, −4.133706736839311, −3.404945532140216, −2.981838325020932, −2.456366763384960, −1.928263804356599, −1.029027431223541, 0, 0, 1.029027431223541, 1.928263804356599, 2.456366763384960, 2.981838325020932, 3.404945532140216, 4.133706736839311, 4.792540375346400, 4.981523469925285, 5.604538326791249, 6.189012896429537, 6.729900099265949, 7.312875713787034, 7.506247171927843, 8.198833490629829, 8.580704235254225, 8.989522494530692, 9.746110466030916, 9.985208662988923, 10.51815495008167, 11.06868988388294, 11.33981216696942, 11.80248412184591, 12.48131283682378, 12.93650780613424, 13.17950630000705

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