Properties

Label 2-1008-1.1-c1-0-8
Degree $2$
Conductor $1008$
Sign $1$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.46·5-s − 7-s + 3.46·11-s + 2·13-s − 3.46·17-s + 4·19-s − 3.46·23-s + 6.99·25-s + 4·31-s − 3.46·35-s + 2·37-s − 10.3·41-s + 4·43-s + 6.92·47-s + 49-s + 6.92·53-s + 11.9·55-s − 6.92·59-s − 10·61-s + 6.92·65-s + 4·67-s − 10.3·71-s + 14·73-s − 3.46·77-s − 8·79-s − 11.9·85-s + 3.46·89-s + ⋯
L(s)  = 1  + 1.54·5-s − 0.377·7-s + 1.04·11-s + 0.554·13-s − 0.840·17-s + 0.917·19-s − 0.722·23-s + 1.39·25-s + 0.718·31-s − 0.585·35-s + 0.328·37-s − 1.62·41-s + 0.609·43-s + 1.01·47-s + 0.142·49-s + 0.951·53-s + 1.61·55-s − 0.901·59-s − 1.28·61-s + 0.859·65-s + 0.488·67-s − 1.23·71-s + 1.63·73-s − 0.394·77-s − 0.900·79-s − 1.30·85-s + 0.367·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.176299601\)
\(L(\frac12)\) \(\approx\) \(2.176299601\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 3.46T + 5T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 3.46T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−9.878203847563377562812830806227, −9.225061506875639872072517656006, −8.608042934606928847597979763575, −7.26340987061140168338190660687, −6.31801403924208100343795722786, −5.94662763023710469644527806871, −4.80292671212293094589886012623, −3.62149936930222138255965531431, −2.39070073068384538551512180901, −1.30751145432054408501031312986, 1.30751145432054408501031312986, 2.39070073068384538551512180901, 3.62149936930222138255965531431, 4.80292671212293094589886012623, 5.94662763023710469644527806871, 6.31801403924208100343795722786, 7.26340987061140168338190660687, 8.608042934606928847597979763575, 9.225061506875639872072517656006, 9.878203847563377562812830806227

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