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 + 0.413·5-s + 2.75·7-s + 9-s − 5.40·11-s + 2.14·13-s − 0.413·15-s + 5.24·17-s − 3.94·19-s − 2.75·21-s + 23-s − 4.82·25-s − 27-s − 29-s + 4.35·31-s + 5.40·33-s + 1.13·35-s − 2.72·37-s − 2.14·39-s − 6.03·41-s − 3.64·43-s + 0.413·45-s + 2.47·47-s + 0.566·49-s − 5.24·51-s − 7.66·53-s − 2.23·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.184·5-s + 1.03·7-s + 0.333·9-s − 1.63·11-s + 0.596·13-s − 0.106·15-s + 1.27·17-s − 0.904·19-s − 0.600·21-s + 0.208·23-s − 0.965·25-s − 0.192·27-s − 0.185·29-s + 0.781·31-s + 0.941·33-s + 0.192·35-s − 0.447·37-s − 0.344·39-s − 0.942·41-s − 0.555·43-s + 0.0616·45-s + 0.361·47-s + 0.0809·49-s − 0.735·51-s − 1.05·53-s − 0.301·55-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 \( 1 - 0.413T + 5T^{2} \)
7 \( 1 - 2.75T + 7T^{2} \)
11 \( 1 + 5.40T + 11T^{2} \)
13 \( 1 - 2.14T + 13T^{2} \)
17 \( 1 - 5.24T + 17T^{2} \)
19 \( 1 + 3.94T + 19T^{2} \)
31 \( 1 - 4.35T + 31T^{2} \)
37 \( 1 + 2.72T + 37T^{2} \)
41 \( 1 + 6.03T + 41T^{2} \)
43 \( 1 + 3.64T + 43T^{2} \)
47 \( 1 - 2.47T + 47T^{2} \)
53 \( 1 + 7.66T + 53T^{2} \)
59 \( 1 - 7.67T + 59T^{2} \)
61 \( 1 + 6.93T + 61T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 + 0.981T + 71T^{2} \)
73 \( 1 + 6.83T + 73T^{2} \)
79 \( 1 - 3.02T + 79T^{2} \)
83 \( 1 + 0.833T + 83T^{2} \)
89 \( 1 + 5.00T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−7.65998439827634211438073859869, −6.76167342882776468137107902322, −5.94412554977968986306567355591, −5.34391435129426568098409944105, −4.90408682429261643942267698943, −4.02606574885553380819981914614, −3.07118895983628251771464566684, −2.09500500187687788701893460618, −1.28718439210837633325223689646, 0, 1.28718439210837633325223689646, 2.09500500187687788701893460618, 3.07118895983628251771464566684, 4.02606574885553380819981914614, 4.90408682429261643942267698943, 5.34391435129426568098409944105, 5.94412554977968986306567355591, 6.76167342882776468137107902322, 7.65998439827634211438073859869

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