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 − 2.66·5-s − 2.40·7-s + 9-s − 0.449·11-s − 7.00·13-s + 2.66·15-s − 4.21·17-s + 5.92·19-s + 2.40·21-s + 23-s + 2.12·25-s − 27-s + 29-s + 1.72·31-s + 0.449·33-s + 6.42·35-s − 10.3·37-s + 7.00·39-s − 2.78·41-s − 3.55·43-s − 2.66·45-s − 10.5·47-s − 1.20·49-s + 4.21·51-s − 14.2·53-s + 1.20·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.19·5-s − 0.909·7-s + 0.333·9-s − 0.135·11-s − 1.94·13-s + 0.689·15-s − 1.02·17-s + 1.36·19-s + 0.525·21-s + 0.208·23-s + 0.424·25-s − 0.192·27-s + 0.185·29-s + 0.309·31-s + 0.0783·33-s + 1.08·35-s − 1.70·37-s + 1.12·39-s − 0.434·41-s − 0.542·43-s − 0.397·45-s − 1.54·47-s − 0.172·49-s + 0.590·51-s − 1.95·53-s + 0.161·55-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

\(L(1)\) \(\approx\) \(0.006605484834\)
\(L(\frac12)\) \(\approx\) \(0.006605484834\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.66T + 5T^{2} \)
7 \( 1 + 2.40T + 7T^{2} \)
11 \( 1 + 0.449T + 11T^{2} \)
13 \( 1 + 7.00T + 13T^{2} \)
17 \( 1 + 4.21T + 17T^{2} \)
19 \( 1 - 5.92T + 19T^{2} \)
31 \( 1 - 1.72T + 31T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 + 2.78T + 41T^{2} \)
43 \( 1 + 3.55T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + 14.2T + 53T^{2} \)
59 \( 1 + 2.05T + 59T^{2} \)
61 \( 1 + 14.9T + 61T^{2} \)
67 \( 1 + 7.71T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 5.88T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 + 6.99T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−7.66758936371704311433609876129, −7.08542018607114454944261249166, −6.69223284940513574882792085023, −5.72887537858103011957830592056, −4.75010857044792658995478117022, −4.62014283776755277260554000460, −3.33875978505122326180148767856, −2.99037270686557741117859167269, −1.68155863113856494680927219822, −0.03987512946949730057391183493, 0.03987512946949730057391183493, 1.68155863113856494680927219822, 2.99037270686557741117859167269, 3.33875978505122326180148767856, 4.62014283776755277260554000460, 4.75010857044792658995478117022, 5.72887537858103011957830592056, 6.69223284940513574882792085023, 7.08542018607114454944261249166, 7.66758936371704311433609876129

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