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 + 0.144·5-s − 2.05·7-s + 9-s − 5.70·11-s − 4.46·13-s − 0.144·15-s + 4.26·17-s + 3.58·19-s + 2.05·21-s + 23-s − 4.97·25-s − 27-s + 29-s − 6.14·31-s + 5.70·33-s − 0.297·35-s + 9.53·37-s + 4.46·39-s − 7.46·41-s − 2.12·43-s + 0.144·45-s − 12.1·47-s − 2.77·49-s − 4.26·51-s + 6.19·53-s − 0.826·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.0647·5-s − 0.776·7-s + 0.333·9-s − 1.72·11-s − 1.23·13-s − 0.0374·15-s + 1.03·17-s + 0.821·19-s + 0.448·21-s + 0.208·23-s − 0.995·25-s − 0.192·27-s + 0.185·29-s − 1.10·31-s + 0.993·33-s − 0.0503·35-s + 1.56·37-s + 0.715·39-s − 1.16·41-s − 0.323·43-s + 0.0215·45-s − 1.76·47-s − 0.396·49-s − 0.596·51-s + 0.850·53-s − 0.111·55-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

\(L(1)\) \(\approx\) \(0.5913961051\)
\(L(\frac12)\) \(\approx\) \(0.5913961051\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.144T + 5T^{2} \)
7 \( 1 + 2.05T + 7T^{2} \)
11 \( 1 + 5.70T + 11T^{2} \)
13 \( 1 + 4.46T + 13T^{2} \)
17 \( 1 - 4.26T + 17T^{2} \)
19 \( 1 - 3.58T + 19T^{2} \)
31 \( 1 + 6.14T + 31T^{2} \)
37 \( 1 - 9.53T + 37T^{2} \)
41 \( 1 + 7.46T + 41T^{2} \)
43 \( 1 + 2.12T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 - 6.19T + 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 - 5.14T + 61T^{2} \)
67 \( 1 + 0.811T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 13.7T + 73T^{2} \)
79 \( 1 - 1.51T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 - 4.10T + 89T^{2} \)
97 \( 1 + 4.29T + 97T^{2} \)
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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

−7.67656921490044866822510316224, −7.26659229124331516510529499165, −6.41147799280412090841445421696, −5.53907754792289670439941034459, −5.27877071064723121845306008174, −4.46917435624086888915482172900, −3.31269419707289418913973949515, −2.81883826424261818423934646007, −1.77073283599473727817972811211, −0.37501078171997851013496673400, 0.37501078171997851013496673400, 1.77073283599473727817972811211, 2.81883826424261818423934646007, 3.31269419707289418913973949515, 4.46917435624086888915482172900, 5.27877071064723121845306008174, 5.53907754792289670439941034459, 6.41147799280412090841445421696, 7.26659229124331516510529499165, 7.67656921490044866822510316224

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