Properties

Degree $2$
Conductor $1008$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s − 2·13-s − 6·17-s + 4·19-s − 4·23-s − 25-s − 6·29-s + 8·31-s − 2·35-s − 10·37-s + 10·41-s − 12·43-s − 8·47-s + 49-s − 6·53-s + 4·59-s − 10·61-s + 4·65-s − 12·67-s + 4·71-s + 2·73-s − 8·79-s + 4·83-s + 12·85-s − 6·89-s − 2·91-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.338·35-s − 1.64·37-s + 1.56·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.520·59-s − 1.28·61-s + 0.496·65-s − 1.46·67-s + 0.474·71-s + 0.234·73-s − 0.900·79-s + 0.439·83-s + 1.30·85-s − 0.635·89-s − 0.209·91-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{1008} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.542335673006963113033744072480, −8.614680372438593469438231251008, −7.86152616392232705724947299443, −7.17945773905277725475919160716, −6.19452637109873977406389525575, −4.99862586128493531522947252735, −4.26888101159701758252770076566, −3.21569388283931405752209731146, −1.87750590194886957605795356311, 0, 1.87750590194886957605795356311, 3.21569388283931405752209731146, 4.26888101159701758252770076566, 4.99862586128493531522947252735, 6.19452637109873977406389525575, 7.17945773905277725475919160716, 7.86152616392232705724947299443, 8.614680372438593469438231251008, 9.542335673006963113033744072480

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