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Properties

Label 2-262080-1.1-c1-0-71
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 − 6·29-s − 4·31-s + 35-s − 8·37-s + 2·41-s − 6·43-s − 8·47-s + 49-s + 12·53-s + 4·55-s + 6·59-s + 6·61-s − 65-s − 16·67-s − 12·71-s − 14·73-s + 4·77-s + 4·79-s − 4·83-s + 6·85-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 − 1.11·29-s − 0.718·31-s + 0.169·35-s − 1.31·37-s + 0.312·41-s − 0.914·43-s − 1.16·47-s + 1/7·49-s + 1.64·53-s + 0.539·55-s + 0.781·59-s + 0.768·61-s − 0.124·65-s − 1.95·67-s − 1.42·71-s − 1.63·73-s + 0.455·77-s + 0.450·79-s − 0.439·83-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: \(1\)
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 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.11537381847690, −12.70602263738049, −12.04382055070032, −11.64972810647878, −11.22539181978918, −10.72737965417645, −10.21873943387755, −10.02917873832199, −9.225428491553839, −8.670843639539034, −8.599777355814953, −7.845576458242645, −7.313762698259343, −7.098472960286677, −6.402416775456974, −5.887289973112071, −5.423650695793132, −4.876666787718288, −4.313016969380629, −3.819154933430257, −3.273563352568310, −2.728762229531739, −2.016658521316778, −1.676039273448088, −0.4754900413016994, 0, 0.4754900413016994, 1.676039273448088, 2.016658521316778, 2.728762229531739, 3.273563352568310, 3.819154933430257, 4.313016969380629, 4.876666787718288, 5.423650695793132, 5.887289973112071, 6.402416775456974, 7.098472960286677, 7.313762698259343, 7.845576458242645, 8.599777355814953, 8.670843639539034, 9.225428491553839, 10.02917873832199, 10.21873943387755, 10.72737965417645, 11.22539181978918, 11.64972810647878, 12.04382055070032, 12.70602263738049, 13.11537381847690

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