Properties

Label 2-1008-1.1-c5-0-9
Degree $2$
Conductor $1008$
Sign $1$
Analytic cond. $161.666$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 36.8·5-s + 49·7-s + 266.·11-s − 1.05e3·13-s − 1.25e3·17-s − 1.28e3·19-s − 982.·23-s − 1.76e3·25-s + 3.19e3·29-s + 1.61e3·31-s − 1.80e3·35-s + 2.13e3·37-s + 8.93e3·41-s + 1.47e4·43-s + 6.41e3·47-s + 2.40e3·49-s − 3.69e4·53-s − 9.80e3·55-s − 2.95e4·59-s + 3.19e4·61-s + 3.87e4·65-s + 2.90e4·67-s − 4.99e4·71-s − 1.25e4·73-s + 1.30e4·77-s − 2.82e4·79-s − 1.10e5·83-s + ⋯
L(s)  = 1  − 0.658·5-s + 0.377·7-s + 0.663·11-s − 1.72·13-s − 1.05·17-s − 0.816·19-s − 0.387·23-s − 0.565·25-s + 0.706·29-s + 0.302·31-s − 0.249·35-s + 0.256·37-s + 0.829·41-s + 1.21·43-s + 0.423·47-s + 0.142·49-s − 1.80·53-s − 0.436·55-s − 1.10·59-s + 1.10·61-s + 1.13·65-s + 0.789·67-s − 1.17·71-s − 0.275·73-s + 0.250·77-s − 0.508·79-s − 1.76·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(161.666\)
Root analytic conductor: \(12.7148\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.137520157\)
\(L(\frac12)\) \(\approx\) \(1.137520157\)
\(L(\frac{7}{2})\) 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 \( 1 - 49T \)
good5 \( 1 + 36.8T + 3.12e3T^{2} \)
11 \( 1 - 266.T + 1.61e5T^{2} \)
13 \( 1 + 1.05e3T + 3.71e5T^{2} \)
17 \( 1 + 1.25e3T + 1.41e6T^{2} \)
19 \( 1 + 1.28e3T + 2.47e6T^{2} \)
23 \( 1 + 982.T + 6.43e6T^{2} \)
29 \( 1 - 3.19e3T + 2.05e7T^{2} \)
31 \( 1 - 1.61e3T + 2.86e7T^{2} \)
37 \( 1 - 2.13e3T + 6.93e7T^{2} \)
41 \( 1 - 8.93e3T + 1.15e8T^{2} \)
43 \( 1 - 1.47e4T + 1.47e8T^{2} \)
47 \( 1 - 6.41e3T + 2.29e8T^{2} \)
53 \( 1 + 3.69e4T + 4.18e8T^{2} \)
59 \( 1 + 2.95e4T + 7.14e8T^{2} \)
61 \( 1 - 3.19e4T + 8.44e8T^{2} \)
67 \( 1 - 2.90e4T + 1.35e9T^{2} \)
71 \( 1 + 4.99e4T + 1.80e9T^{2} \)
73 \( 1 + 1.25e4T + 2.07e9T^{2} \)
79 \( 1 + 2.82e4T + 3.07e9T^{2} \)
83 \( 1 + 1.10e5T + 3.93e9T^{2} \)
89 \( 1 + 5.65e4T + 5.58e9T^{2} \)
97 \( 1 + 1.08e5T + 8.58e9T^{2} \)
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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

−9.227162080573616326149463497412, −8.339874941391726503982502114342, −7.56913991502309682012181699576, −6.83072288583516736932104934832, −5.85636012996418654477309165653, −4.54064118696689651602383029089, −4.25948222166778759294155156656, −2.80723990094537328423638047064, −1.89047597093555118726986543216, −0.44781218490007675307819611395, 0.44781218490007675307819611395, 1.89047597093555118726986543216, 2.80723990094537328423638047064, 4.25948222166778759294155156656, 4.54064118696689651602383029089, 5.85636012996418654477309165653, 6.83072288583516736932104934832, 7.56913991502309682012181699576, 8.339874941391726503982502114342, 9.227162080573616326149463497412

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