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 − 1.83·5-s + 1.71·7-s + 9-s − 0.324·11-s + 2.24·13-s + 1.83·15-s + 3.03·17-s − 6.93·19-s − 1.71·21-s + 23-s − 1.63·25-s − 27-s + 29-s + 0.153·31-s + 0.324·33-s − 3.14·35-s + 7.96·37-s − 2.24·39-s + 5.42·41-s + 4.81·43-s − 1.83·45-s − 4.18·47-s − 4.06·49-s − 3.03·51-s + 11.5·53-s + 0.595·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.820·5-s + 0.647·7-s + 0.333·9-s − 0.0978·11-s + 0.621·13-s + 0.473·15-s + 0.735·17-s − 1.59·19-s − 0.373·21-s + 0.208·23-s − 0.326·25-s − 0.192·27-s + 0.185·29-s + 0.0275·31-s + 0.0565·33-s − 0.531·35-s + 1.30·37-s − 0.358·39-s + 0.847·41-s + 0.733·43-s − 0.273·45-s − 0.611·47-s − 0.580·49-s − 0.424·51-s + 1.58·53-s + 0.0803·55-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

\(L(1)\) \(\approx\) \(1.321397779\)
\(L(\frac12)\) \(\approx\) \(1.321397779\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.83T + 5T^{2} \)
7 \( 1 - 1.71T + 7T^{2} \)
11 \( 1 + 0.324T + 11T^{2} \)
13 \( 1 - 2.24T + 13T^{2} \)
17 \( 1 - 3.03T + 17T^{2} \)
19 \( 1 + 6.93T + 19T^{2} \)
31 \( 1 - 0.153T + 31T^{2} \)
37 \( 1 - 7.96T + 37T^{2} \)
41 \( 1 - 5.42T + 41T^{2} \)
43 \( 1 - 4.81T + 43T^{2} \)
47 \( 1 + 4.18T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 2.10T + 59T^{2} \)
61 \( 1 + 1.10T + 61T^{2} \)
67 \( 1 + 3.92T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 + 6.02T + 73T^{2} \)
79 \( 1 + 9.02T + 79T^{2} \)
83 \( 1 + 0.162T + 83T^{2} \)
89 \( 1 - 9.59T + 89T^{2} \)
97 \( 1 + 1.19T + 97T^{2} \)
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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

−7.69370567734486943079249312300, −7.36142494950250779991095149620, −6.23029431189245307658584572454, −5.94508276246278216451232625839, −4.90731034706536141440712071751, −4.31713651026995694887870278953, −3.74668506395560994129075328609, −2.67171017807679240282518946206, −1.61688535608208590588788063930, −0.60568839191446078091643253597, 0.60568839191446078091643253597, 1.61688535608208590588788063930, 2.67171017807679240282518946206, 3.74668506395560994129075328609, 4.31713651026995694887870278953, 4.90731034706536141440712071751, 5.94508276246278216451232625839, 6.23029431189245307658584572454, 7.36142494950250779991095149620, 7.69370567734486943079249312300

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