Properties

Degree $2$
Conductor $8004$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4.28·5-s + 1.70·7-s + 9-s + 4.51·11-s − 1.48·13-s + 4.28·15-s − 1.55·17-s − 1.31·19-s − 1.70·21-s + 23-s + 13.3·25-s − 27-s + 29-s − 2.61·31-s − 4.51·33-s − 7.29·35-s − 0.322·37-s + 1.48·39-s − 7.70·41-s + 4.70·43-s − 4.28·45-s + 13.0·47-s − 4.10·49-s + 1.55·51-s + 7.24·53-s − 19.3·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.91·5-s + 0.642·7-s + 0.333·9-s + 1.36·11-s − 0.412·13-s + 1.10·15-s − 0.376·17-s − 0.301·19-s − 0.371·21-s + 0.208·23-s + 2.67·25-s − 0.192·27-s + 0.185·29-s − 0.469·31-s − 0.785·33-s − 1.23·35-s − 0.0530·37-s + 0.238·39-s − 1.20·41-s + 0.718·43-s − 0.638·45-s + 1.90·47-s − 0.586·49-s + 0.217·51-s + 0.994·53-s − 2.60·55-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: \(0\)
Selberg data: \((2,\ 8004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.006384207\)
\(L(\frac12)\) \(\approx\) \(1.006384207\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.28T + 5T^{2} \)
7 \( 1 - 1.70T + 7T^{2} \)
11 \( 1 - 4.51T + 11T^{2} \)
13 \( 1 + 1.48T + 13T^{2} \)
17 \( 1 + 1.55T + 17T^{2} \)
19 \( 1 + 1.31T + 19T^{2} \)
31 \( 1 + 2.61T + 31T^{2} \)
37 \( 1 + 0.322T + 37T^{2} \)
41 \( 1 + 7.70T + 41T^{2} \)
43 \( 1 - 4.70T + 43T^{2} \)
47 \( 1 - 13.0T + 47T^{2} \)
53 \( 1 - 7.24T + 53T^{2} \)
59 \( 1 - 9.70T + 59T^{2} \)
61 \( 1 + 4.23T + 61T^{2} \)
67 \( 1 + 3.71T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 + 17.6T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
97 \( 1 + 0.650T + 97T^{2} \)
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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

−7.74560332721935459618045330355, −7.05207749887601380428763030319, −6.78584603381306513292100487659, −5.68020585766135465591887864334, −4.86315121877878091528099386020, −4.12647226733833772928882214168, −3.92432533954861396745645381026, −2.80878813398448831275471788344, −1.50363222793748032125586744593, −0.54370007951976906065920448198, 0.54370007951976906065920448198, 1.50363222793748032125586744593, 2.80878813398448831275471788344, 3.92432533954861396745645381026, 4.12647226733833772928882214168, 4.86315121877878091528099386020, 5.68020585766135465591887864334, 6.78584603381306513292100487659, 7.05207749887601380428763030319, 7.74560332721935459618045330355

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