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 + 0.282·5-s + 5.02·7-s + 9-s + 1.61·11-s + 0.572·13-s − 0.282·15-s − 0.464·17-s + 0.460·19-s − 5.02·21-s + 23-s − 4.92·25-s − 27-s + 29-s − 1.38·31-s − 1.61·33-s + 1.41·35-s + 4.37·37-s − 0.572·39-s + 2.95·41-s − 6.06·43-s + 0.282·45-s + 6.36·47-s + 18.2·49-s + 0.464·51-s − 5.96·53-s + 0.456·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.126·5-s + 1.89·7-s + 0.333·9-s + 0.487·11-s + 0.158·13-s − 0.0729·15-s − 0.112·17-s + 0.105·19-s − 1.09·21-s + 0.208·23-s − 0.984·25-s − 0.192·27-s + 0.185·29-s − 0.248·31-s − 0.281·33-s + 0.239·35-s + 0.719·37-s − 0.0916·39-s + 0.461·41-s − 0.925·43-s + 0.0420·45-s + 0.928·47-s + 2.60·49-s + 0.0650·51-s − 0.818·53-s + 0.0615·55-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

\(L(1)\) \(\approx\) \(2.434670790\)
\(L(\frac12)\) \(\approx\) \(2.434670790\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.282T + 5T^{2} \)
7 \( 1 - 5.02T + 7T^{2} \)
11 \( 1 - 1.61T + 11T^{2} \)
13 \( 1 - 0.572T + 13T^{2} \)
17 \( 1 + 0.464T + 17T^{2} \)
19 \( 1 - 0.460T + 19T^{2} \)
31 \( 1 + 1.38T + 31T^{2} \)
37 \( 1 - 4.37T + 37T^{2} \)
41 \( 1 - 2.95T + 41T^{2} \)
43 \( 1 + 6.06T + 43T^{2} \)
47 \( 1 - 6.36T + 47T^{2} \)
53 \( 1 + 5.96T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 7.82T + 61T^{2} \)
67 \( 1 + 4.23T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 - 5.71T + 73T^{2} \)
79 \( 1 + 5.81T + 79T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 - 5.33T + 89T^{2} \)
97 \( 1 + 17.5T + 97T^{2} \)
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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

−7.947716798021791226828222394960, −7.14516657865401313616012051713, −6.45040640234231467390411118923, −5.55298994957442124376194549414, −5.16041472368249485950821054338, −4.34333046326147943267345310469, −3.80403542094274885158436998849, −2.41556864494185527712893675926, −1.66877932542099159593341759602, −0.855558573664125524571347207579, 0.855558573664125524571347207579, 1.66877932542099159593341759602, 2.41556864494185527712893675926, 3.80403542094274885158436998849, 4.34333046326147943267345310469, 5.16041472368249485950821054338, 5.55298994957442124376194549414, 6.45040640234231467390411118923, 7.14516657865401313616012051713, 7.947716798021791226828222394960

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