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 + 1.42·5-s − 1.49·7-s + 9-s + 4.45·11-s − 0.159·13-s − 1.42·15-s + 3.47·17-s + 1.29·19-s + 1.49·21-s + 23-s − 2.96·25-s − 27-s − 29-s − 8.87·31-s − 4.45·33-s − 2.12·35-s − 9.07·37-s + 0.159·39-s − 6.40·41-s − 12.0·43-s + 1.42·45-s − 10.2·47-s − 4.77·49-s − 3.47·51-s + 8.00·53-s + 6.35·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.637·5-s − 0.563·7-s + 0.333·9-s + 1.34·11-s − 0.0440·13-s − 0.368·15-s + 0.842·17-s + 0.297·19-s + 0.325·21-s + 0.208·23-s − 0.593·25-s − 0.192·27-s − 0.185·29-s − 1.59·31-s − 0.776·33-s − 0.359·35-s − 1.49·37-s + 0.0254·39-s − 1.00·41-s − 1.83·43-s + 0.212·45-s − 1.49·47-s − 0.682·49-s − 0.486·51-s + 1.09·53-s + 0.857·55-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 \( 1 - 1.42T + 5T^{2} \)
7 \( 1 + 1.49T + 7T^{2} \)
11 \( 1 - 4.45T + 11T^{2} \)
13 \( 1 + 0.159T + 13T^{2} \)
17 \( 1 - 3.47T + 17T^{2} \)
19 \( 1 - 1.29T + 19T^{2} \)
31 \( 1 + 8.87T + 31T^{2} \)
37 \( 1 + 9.07T + 37T^{2} \)
41 \( 1 + 6.40T + 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 8.00T + 53T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 + 0.485T + 67T^{2} \)
71 \( 1 - 7.64T + 71T^{2} \)
73 \( 1 + 3.71T + 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 - 1.30T + 83T^{2} \)
89 \( 1 + 18.2T + 89T^{2} \)
97 \( 1 - 2.78T + 97T^{2} \)
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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

−7.19770457294778528086093844358, −6.75999072415147620276881481363, −6.13126693306485412386717421128, −5.43789878113671765188587451018, −4.89358811047497625707742084228, −3.63435970841590504283821934425, −3.44177881151764552085067872042, −1.95649843065576687548074786143, −1.35485314070441286542663322758, 0, 1.35485314070441286542663322758, 1.95649843065576687548074786143, 3.44177881151764552085067872042, 3.63435970841590504283821934425, 4.89358811047497625707742084228, 5.43789878113671765188587451018, 6.13126693306485412386717421128, 6.75999072415147620276881481363, 7.19770457294778528086093844358

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