Properties

Degree $2$
Conductor $8004$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3.48·5-s + 2.18·7-s + 9-s − 4.51·11-s + 4.71·13-s + 3.48·15-s − 4.53·17-s − 1.79·19-s − 2.18·21-s − 23-s + 7.13·25-s − 27-s + 29-s + 5.46·31-s + 4.51·33-s − 7.61·35-s + 2.75·37-s − 4.71·39-s + 0.252·41-s + 5.62·43-s − 3.48·45-s + 2.83·47-s − 2.22·49-s + 4.53·51-s − 4.43·53-s + 15.7·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.55·5-s + 0.825·7-s + 0.333·9-s − 1.36·11-s + 1.30·13-s + 0.899·15-s − 1.09·17-s − 0.412·19-s − 0.476·21-s − 0.208·23-s + 1.42·25-s − 0.192·27-s + 0.185·29-s + 0.981·31-s + 0.785·33-s − 1.28·35-s + 0.452·37-s − 0.755·39-s + 0.0394·41-s + 0.857·43-s − 0.519·45-s + 0.413·47-s − 0.318·49-s + 0.634·51-s − 0.609·53-s + 2.11·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8004\)    =    \(2^{2} \cdot 3 \cdot 23 \cdot 29\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{8004} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8004,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 + 3.48T + 5T^{2} \)
7 \( 1 - 2.18T + 7T^{2} \)
11 \( 1 + 4.51T + 11T^{2} \)
13 \( 1 - 4.71T + 13T^{2} \)
17 \( 1 + 4.53T + 17T^{2} \)
19 \( 1 + 1.79T + 19T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 - 2.75T + 37T^{2} \)
41 \( 1 - 0.252T + 41T^{2} \)
43 \( 1 - 5.62T + 43T^{2} \)
47 \( 1 - 2.83T + 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 - 4.10T + 59T^{2} \)
61 \( 1 + 0.160T + 61T^{2} \)
67 \( 1 + 3.72T + 67T^{2} \)
71 \( 1 + 0.134T + 71T^{2} \)
73 \( 1 - 8.61T + 73T^{2} \)
79 \( 1 + 2.48T + 79T^{2} \)
83 \( 1 + 4.04T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−7.62304184787808078648483105266, −6.85247563347944702894070696904, −6.12821215873981913378135716602, −5.29550292678438082654165207102, −4.49188279144722809352455622178, −4.19264800132449119012527582601, −3.21295244883324497717281458323, −2.23370913571935563586287951206, −0.992089007376262196524023425623, 0, 0.992089007376262196524023425623, 2.23370913571935563586287951206, 3.21295244883324497717281458323, 4.19264800132449119012527582601, 4.49188279144722809352455622178, 5.29550292678438082654165207102, 6.12821215873981913378135716602, 6.85247563347944702894070696904, 7.62304184787808078648483105266

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