Properties

Label 2-1008-1.1-c5-0-10
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  − 60.4·5-s − 49·7-s + 223.·11-s + 250.·13-s + 421.·17-s − 1.10e3·19-s − 825.·23-s + 526.·25-s + 108.·29-s − 619.·31-s + 2.96e3·35-s + 2.91e3·37-s − 5.97e3·41-s − 1.15e4·43-s − 1.58e3·47-s + 2.40e3·49-s − 1.87e4·53-s − 1.34e4·55-s + 5.58e3·59-s − 1.80e4·61-s − 1.51e4·65-s − 4.35e4·67-s + 3.17e4·71-s − 6.70e4·73-s − 1.09e4·77-s + 573.·79-s + 1.22e5·83-s + ⋯
L(s)  = 1  − 1.08·5-s − 0.377·7-s + 0.556·11-s + 0.411·13-s + 0.353·17-s − 0.704·19-s − 0.325·23-s + 0.168·25-s + 0.0240·29-s − 0.115·31-s + 0.408·35-s + 0.350·37-s − 0.555·41-s − 0.956·43-s − 0.104·47-s + 0.142·49-s − 0.917·53-s − 0.601·55-s + 0.208·59-s − 0.622·61-s − 0.445·65-s − 1.18·67-s + 0.746·71-s − 1.47·73-s − 0.210·77-s + 0.0103·79-s + 1.95·83-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

\(L(3)\) \(\approx\) \(1.154485881\)
\(L(\frac12)\) \(\approx\) \(1.154485881\)
\(L(\frac{7}{2})\) 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 \( 1 + 60.4T + 3.12e3T^{2} \)
11 \( 1 - 223.T + 1.61e5T^{2} \)
13 \( 1 - 250.T + 3.71e5T^{2} \)
17 \( 1 - 421.T + 1.41e6T^{2} \)
19 \( 1 + 1.10e3T + 2.47e6T^{2} \)
23 \( 1 + 825.T + 6.43e6T^{2} \)
29 \( 1 - 108.T + 2.05e7T^{2} \)
31 \( 1 + 619.T + 2.86e7T^{2} \)
37 \( 1 - 2.91e3T + 6.93e7T^{2} \)
41 \( 1 + 5.97e3T + 1.15e8T^{2} \)
43 \( 1 + 1.15e4T + 1.47e8T^{2} \)
47 \( 1 + 1.58e3T + 2.29e8T^{2} \)
53 \( 1 + 1.87e4T + 4.18e8T^{2} \)
59 \( 1 - 5.58e3T + 7.14e8T^{2} \)
61 \( 1 + 1.80e4T + 8.44e8T^{2} \)
67 \( 1 + 4.35e4T + 1.35e9T^{2} \)
71 \( 1 - 3.17e4T + 1.80e9T^{2} \)
73 \( 1 + 6.70e4T + 2.07e9T^{2} \)
79 \( 1 - 573.T + 3.07e9T^{2} \)
83 \( 1 - 1.22e5T + 3.93e9T^{2} \)
89 \( 1 - 3.37e4T + 5.58e9T^{2} \)
97 \( 1 + 1.29e5T + 8.58e9T^{2} \)
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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

−9.143425804005610720596668849924, −8.341978885992136615692489784352, −7.64386010879464661436241738590, −6.71617997821547441167861923240, −5.95354460019274356443735578834, −4.69745102408123532946109220437, −3.86973685056225665922856852398, −3.14484314405418520022129797794, −1.72296216793397064908415001792, −0.46722074126278856589504561030, 0.46722074126278856589504561030, 1.72296216793397064908415001792, 3.14484314405418520022129797794, 3.86973685056225665922856852398, 4.69745102408123532946109220437, 5.95354460019274356443735578834, 6.71617997821547441167861923240, 7.64386010879464661436241738590, 8.341978885992136615692489784352, 9.143425804005610720596668849924

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