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 + 1.38·5-s − 3.43·7-s + 9-s + 2.64·11-s + 3.17·13-s − 1.38·15-s − 7.44·17-s + 7.20·19-s + 3.43·21-s + 23-s − 3.07·25-s − 27-s + 29-s − 6.70·31-s − 2.64·33-s − 4.76·35-s + 2.03·37-s − 3.17·39-s + 7.14·41-s − 3.08·43-s + 1.38·45-s + 5.04·47-s + 4.80·49-s + 7.44·51-s + 9.24·53-s + 3.66·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.619·5-s − 1.29·7-s + 0.333·9-s + 0.798·11-s + 0.881·13-s − 0.357·15-s − 1.80·17-s + 1.65·19-s + 0.749·21-s + 0.208·23-s − 0.615·25-s − 0.192·27-s + 0.185·29-s − 1.20·31-s − 0.460·33-s − 0.804·35-s + 0.334·37-s − 0.509·39-s + 1.11·41-s − 0.470·43-s + 0.206·45-s + 0.736·47-s + 0.686·49-s + 1.04·51-s + 1.27·53-s + 0.494·55-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

\(L(1)\) \(\approx\) \(1.519055093\)
\(L(\frac12)\) \(\approx\) \(1.519055093\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.38T + 5T^{2} \)
7 \( 1 + 3.43T + 7T^{2} \)
11 \( 1 - 2.64T + 11T^{2} \)
13 \( 1 - 3.17T + 13T^{2} \)
17 \( 1 + 7.44T + 17T^{2} \)
19 \( 1 - 7.20T + 19T^{2} \)
31 \( 1 + 6.70T + 31T^{2} \)
37 \( 1 - 2.03T + 37T^{2} \)
41 \( 1 - 7.14T + 41T^{2} \)
43 \( 1 + 3.08T + 43T^{2} \)
47 \( 1 - 5.04T + 47T^{2} \)
53 \( 1 - 9.24T + 53T^{2} \)
59 \( 1 - 0.888T + 59T^{2} \)
61 \( 1 + 0.116T + 61T^{2} \)
67 \( 1 + 7.88T + 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 - 8.68T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 2.69T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 - 8.50T + 97T^{2} \)
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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

−7.61055490384054859556912306253, −6.94259986219880366367031241934, −6.35797485903441638608344471063, −5.93071996588580335368383774840, −5.22556438623135775601690725932, −4.17212616081814113361132371130, −3.62088154794836981244158807587, −2.68185551038203717761494357737, −1.67758382359400430339514539404, −0.63752559154714647095328542268, 0.63752559154714647095328542268, 1.67758382359400430339514539404, 2.68185551038203717761494357737, 3.62088154794836981244158807587, 4.17212616081814113361132371130, 5.22556438623135775601690725932, 5.93071996588580335368383774840, 6.35797485903441638608344471063, 6.94259986219880366367031241934, 7.61055490384054859556912306253

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