Properties

Label 2-1008-1.1-c5-0-53
Degree $2$
Conductor $1008$
Sign $-1$
Analytic cond. $161.666$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 26·5-s + 49·7-s − 470·11-s + 642·13-s + 1.01e3·17-s − 1.53e3·19-s + 430·23-s − 2.44e3·25-s + 6.73e3·29-s − 2.26e3·31-s − 1.27e3·35-s − 9.57e3·37-s + 1.44e4·41-s + 9.74e3·43-s − 1.70e4·47-s + 2.40e3·49-s + 7.59e3·53-s + 1.22e4·55-s + 1.89e4·59-s − 3.67e4·61-s − 1.66e4·65-s + 3.67e4·67-s − 1.83e4·71-s + 3.63e4·73-s − 2.30e4·77-s − 2.97e4·79-s + 2.82e4·83-s + ⋯
L(s)  = 1  − 0.465·5-s + 0.377·7-s − 1.17·11-s + 1.05·13-s + 0.847·17-s − 0.973·19-s + 0.169·23-s − 0.783·25-s + 1.48·29-s − 0.423·31-s − 0.175·35-s − 1.14·37-s + 1.33·41-s + 0.803·43-s − 1.12·47-s + 1/7·49-s + 0.371·53-s + 0.544·55-s + 0.707·59-s − 1.26·61-s − 0.490·65-s + 1.00·67-s − 0.431·71-s + 0.799·73-s − 0.442·77-s − 0.536·79-s + 0.449·83-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - p^{2} T \)
good5 \( 1 + 26 T + p^{5} T^{2} \)
11 \( 1 + 470 T + p^{5} T^{2} \)
13 \( 1 - 642 T + p^{5} T^{2} \)
17 \( 1 - 1010 T + p^{5} T^{2} \)
19 \( 1 + 1532 T + p^{5} T^{2} \)
23 \( 1 - 430 T + p^{5} T^{2} \)
29 \( 1 - 6736 T + p^{5} T^{2} \)
31 \( 1 + 2268 T + p^{5} T^{2} \)
37 \( 1 + 9574 T + p^{5} T^{2} \)
41 \( 1 - 14406 T + p^{5} T^{2} \)
43 \( 1 - 9748 T + p^{5} T^{2} \)
47 \( 1 + 17004 T + p^{5} T^{2} \)
53 \( 1 - 7596 T + p^{5} T^{2} \)
59 \( 1 - 18908 T + p^{5} T^{2} \)
61 \( 1 + 36762 T + p^{5} T^{2} \)
67 \( 1 - 36788 T + p^{5} T^{2} \)
71 \( 1 + 18326 T + p^{5} T^{2} \)
73 \( 1 - 36382 T + p^{5} T^{2} \)
79 \( 1 + 29784 T + p^{5} T^{2} \)
83 \( 1 - 28240 T + p^{5} T^{2} \)
89 \( 1 + 75954 T + p^{5} T^{2} \)
97 \( 1 + 80690 T + p^{5} T^{2} \)
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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

−8.531387375778994462296587053369, −8.121862419090145728397520537659, −7.27628654553010140376676717869, −6.19015033528154712919915833250, −5.36627080010550989426274725176, −4.38848961380452654392506683264, −3.46130744872734496231112241138, −2.38917839451821546106549035235, −1.16520517387825486182135068894, 0, 1.16520517387825486182135068894, 2.38917839451821546106549035235, 3.46130744872734496231112241138, 4.38848961380452654392506683264, 5.36627080010550989426274725176, 6.19015033528154712919915833250, 7.27628654553010140376676717869, 8.121862419090145728397520537659, 8.531387375778994462296587053369

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