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 + 3.54·5-s + 2.31·7-s + 9-s − 0.0711·11-s − 0.620·13-s − 3.54·15-s − 4.97·17-s − 6.83·19-s − 2.31·21-s + 23-s + 7.53·25-s − 27-s − 29-s − 2.38·31-s + 0.0711·33-s + 8.18·35-s − 7.78·37-s + 0.620·39-s − 6.42·41-s + 1.04·43-s + 3.54·45-s + 5.51·47-s − 1.66·49-s + 4.97·51-s − 5.37·53-s − 0.252·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.58·5-s + 0.873·7-s + 0.333·9-s − 0.0214·11-s − 0.172·13-s − 0.914·15-s − 1.20·17-s − 1.56·19-s − 0.504·21-s + 0.208·23-s + 1.50·25-s − 0.192·27-s − 0.185·29-s − 0.428·31-s + 0.0123·33-s + 1.38·35-s − 1.28·37-s + 0.0994·39-s − 1.00·41-s + 0.159·43-s + 0.527·45-s + 0.804·47-s − 0.237·49-s + 0.697·51-s − 0.738·53-s − 0.0339·55-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 \( 1 - 3.54T + 5T^{2} \)
7 \( 1 - 2.31T + 7T^{2} \)
11 \( 1 + 0.0711T + 11T^{2} \)
13 \( 1 + 0.620T + 13T^{2} \)
17 \( 1 + 4.97T + 17T^{2} \)
19 \( 1 + 6.83T + 19T^{2} \)
31 \( 1 + 2.38T + 31T^{2} \)
37 \( 1 + 7.78T + 37T^{2} \)
41 \( 1 + 6.42T + 41T^{2} \)
43 \( 1 - 1.04T + 43T^{2} \)
47 \( 1 - 5.51T + 47T^{2} \)
53 \( 1 + 5.37T + 53T^{2} \)
59 \( 1 - 0.604T + 59T^{2} \)
61 \( 1 + 9.02T + 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + 3.52T + 71T^{2} \)
73 \( 1 + 4.78T + 73T^{2} \)
79 \( 1 - 1.72T + 79T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 + 8.77T + 89T^{2} \)
97 \( 1 - 5.44T + 97T^{2} \)
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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

−7.26472810657483514097727577465, −6.67263482392236936748134000349, −6.06383475962409763150035617355, −5.46084044284973293403172402658, −4.76620806827764248985572271360, −4.22316973867577100294483297070, −2.87069718912065476508359638693, −1.90451609430070012392623032903, −1.61551138226079428243645151775, 0, 1.61551138226079428243645151775, 1.90451609430070012392623032903, 2.87069718912065476508359638693, 4.22316973867577100294483297070, 4.76620806827764248985572271360, 5.46084044284973293403172402658, 6.06383475962409763150035617355, 6.67263482392236936748134000349, 7.26472810657483514097727577465

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