Properties

Label 2-8002-1.1-c1-0-106
Degree $2$
Conductor $8002$
Sign $-1$
Analytic cond. $63.8962$
Root an. cond. $7.99351$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3.43·3-s + 4-s − 2.28·5-s − 3.43·6-s − 3.85·7-s + 8-s + 8.81·9-s − 2.28·10-s + 0.361·11-s − 3.43·12-s − 4.38·13-s − 3.85·14-s + 7.84·15-s + 16-s + 1.54·17-s + 8.81·18-s − 7.18·19-s − 2.28·20-s + 13.2·21-s + 0.361·22-s + 6.47·23-s − 3.43·24-s + 0.211·25-s − 4.38·26-s − 19.9·27-s − 3.85·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.98·3-s + 0.5·4-s − 1.02·5-s − 1.40·6-s − 1.45·7-s + 0.353·8-s + 2.93·9-s − 0.721·10-s + 0.108·11-s − 0.992·12-s − 1.21·13-s − 1.03·14-s + 2.02·15-s + 0.250·16-s + 0.374·17-s + 2.07·18-s − 1.64·19-s − 0.510·20-s + 2.89·21-s + 0.0769·22-s + 1.35·23-s − 0.701·24-s + 0.0422·25-s − 0.860·26-s − 3.84·27-s − 0.729·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8002\)    =    \(2 \cdot 4001\)
Sign: $-1$
Analytic conductor: \(63.8962\)
Root analytic conductor: \(7.99351\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8002,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
4001 \( 1+O(T) \)
good3 \( 1 + 3.43T + 3T^{2} \)
5 \( 1 + 2.28T + 5T^{2} \)
7 \( 1 + 3.85T + 7T^{2} \)
11 \( 1 - 0.361T + 11T^{2} \)
13 \( 1 + 4.38T + 13T^{2} \)
17 \( 1 - 1.54T + 17T^{2} \)
19 \( 1 + 7.18T + 19T^{2} \)
23 \( 1 - 6.47T + 23T^{2} \)
29 \( 1 + 4.60T + 29T^{2} \)
31 \( 1 - 9.01T + 31T^{2} \)
37 \( 1 + 0.704T + 37T^{2} \)
41 \( 1 + 1.54T + 41T^{2} \)
43 \( 1 - 6.07T + 43T^{2} \)
47 \( 1 - 6.06T + 47T^{2} \)
53 \( 1 - 2.49T + 53T^{2} \)
59 \( 1 - 9.64T + 59T^{2} \)
61 \( 1 + 5.89T + 61T^{2} \)
67 \( 1 - 9.47T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 + 16.0T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 - 18.3T + 89T^{2} \)
97 \( 1 - 1.80T + 97T^{2} \)
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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

−7.09988742403313891948701821339, −6.71324916011434198107316590883, −6.06188725038045797756557141338, −5.46544445762560253585200546206, −4.60077090181906498834046025340, −4.22291491553475578199726539748, −3.41331182809365153328172767286, −2.31910107267963884458754788028, −0.809637002682085496100135516037, 0, 0.809637002682085496100135516037, 2.31910107267963884458754788028, 3.41331182809365153328172767286, 4.22291491553475578199726539748, 4.60077090181906498834046025340, 5.46544445762560253585200546206, 6.06188725038045797756557141338, 6.71324916011434198107316590883, 7.09988742403313891948701821339

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