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 + 4.02·5-s + 4.04·7-s + 9-s − 3.42·11-s + 4.92·13-s − 4.02·15-s + 4.60·17-s + 7.63·19-s − 4.04·21-s + 23-s + 11.1·25-s − 27-s + 29-s − 4.21·31-s + 3.42·33-s + 16.2·35-s + 3.25·37-s − 4.92·39-s − 3.81·41-s − 4.87·43-s + 4.02·45-s + 4.28·47-s + 9.34·49-s − 4.60·51-s + 11.5·53-s − 13.7·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.79·5-s + 1.52·7-s + 0.333·9-s − 1.03·11-s + 1.36·13-s − 1.03·15-s + 1.11·17-s + 1.75·19-s − 0.882·21-s + 0.208·23-s + 2.23·25-s − 0.192·27-s + 0.185·29-s − 0.757·31-s + 0.596·33-s + 2.74·35-s + 0.534·37-s − 0.787·39-s − 0.595·41-s − 0.744·43-s + 0.599·45-s + 0.624·47-s + 1.33·49-s − 0.645·51-s + 1.59·53-s − 1.85·55-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

\(L(1)\) \(\approx\) \(3.626796886\)
\(L(\frac12)\) \(\approx\) \(3.626796886\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.02T + 5T^{2} \)
7 \( 1 - 4.04T + 7T^{2} \)
11 \( 1 + 3.42T + 11T^{2} \)
13 \( 1 - 4.92T + 13T^{2} \)
17 \( 1 - 4.60T + 17T^{2} \)
19 \( 1 - 7.63T + 19T^{2} \)
31 \( 1 + 4.21T + 31T^{2} \)
37 \( 1 - 3.25T + 37T^{2} \)
41 \( 1 + 3.81T + 41T^{2} \)
43 \( 1 + 4.87T + 43T^{2} \)
47 \( 1 - 4.28T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 - 4.70T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + 15.2T + 67T^{2} \)
71 \( 1 + 7.01T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 + 5.09T + 79T^{2} \)
83 \( 1 + 0.476T + 83T^{2} \)
89 \( 1 - 8.93T + 89T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−7.69858652208651648528463479661, −7.24833268111680647110724985910, −6.11597673078849600778084527576, −5.63116485508363675261230632942, −5.29480059734058435585044076329, −4.66060105798846080200767251396, −3.40711390130569566113969744040, −2.51483989527770403602132629746, −1.44035737178079189242228739008, −1.21068728938508431452046113396, 1.21068728938508431452046113396, 1.44035737178079189242228739008, 2.51483989527770403602132629746, 3.40711390130569566113969744040, 4.66060105798846080200767251396, 5.29480059734058435585044076329, 5.63116485508363675261230632942, 6.11597673078849600778084527576, 7.24833268111680647110724985910, 7.69858652208651648528463479661

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