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 − 0.425·5-s − 4.01·7-s + 9-s + 2.74·11-s + 2.75·13-s + 0.425·15-s − 0.750·17-s − 4.08·19-s + 4.01·21-s + 23-s − 4.81·25-s − 27-s − 29-s + 3.50·31-s − 2.74·33-s + 1.70·35-s − 5.25·37-s − 2.75·39-s + 11.5·41-s + 7.71·43-s − 0.425·45-s + 3.88·47-s + 9.14·49-s + 0.750·51-s − 0.561·53-s − 1.16·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.190·5-s − 1.51·7-s + 0.333·9-s + 0.828·11-s + 0.762·13-s + 0.109·15-s − 0.181·17-s − 0.937·19-s + 0.876·21-s + 0.208·23-s − 0.963·25-s − 0.192·27-s − 0.185·29-s + 0.629·31-s − 0.478·33-s + 0.289·35-s − 0.863·37-s − 0.440·39-s + 1.79·41-s + 1.17·43-s − 0.0634·45-s + 0.567·47-s + 1.30·49-s + 0.105·51-s − 0.0771·53-s − 0.157·55-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 \( 1 + 0.425T + 5T^{2} \)
7 \( 1 + 4.01T + 7T^{2} \)
11 \( 1 - 2.74T + 11T^{2} \)
13 \( 1 - 2.75T + 13T^{2} \)
17 \( 1 + 0.750T + 17T^{2} \)
19 \( 1 + 4.08T + 19T^{2} \)
31 \( 1 - 3.50T + 31T^{2} \)
37 \( 1 + 5.25T + 37T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 - 7.71T + 43T^{2} \)
47 \( 1 - 3.88T + 47T^{2} \)
53 \( 1 + 0.561T + 53T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 8.17T + 73T^{2} \)
79 \( 1 + 1.31T + 79T^{2} \)
83 \( 1 - 8.92T + 83T^{2} \)
89 \( 1 - 2.90T + 89T^{2} \)
97 \( 1 + 17.7T + 97T^{2} \)
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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

−7.33156965625842879028396504229, −6.59563785702239970498900260745, −6.17213811706438036885292211333, −5.70818721908347095593197405392, −4.47555183968612881126617922690, −3.94666691804444338041445608259, −3.25918022167152415188277256580, −2.24369331993859501726431566492, −1.04605010374714189542577855076, 0, 1.04605010374714189542577855076, 2.24369331993859501726431566492, 3.25918022167152415188277256580, 3.94666691804444338041445608259, 4.47555183968612881126617922690, 5.70818721908347095593197405392, 6.17213811706438036885292211333, 6.59563785702239970498900260745, 7.33156965625842879028396504229

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