Properties

Label 2-8002-1.1-c1-0-31
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.50·3-s + 4-s − 3.69·5-s + 2.50·6-s − 2.32·7-s − 8-s + 3.29·9-s + 3.69·10-s + 1.21·11-s − 2.50·12-s − 1.51·13-s + 2.32·14-s + 9.26·15-s + 16-s + 4.40·17-s − 3.29·18-s + 1.84·19-s − 3.69·20-s + 5.83·21-s − 1.21·22-s + 3.38·23-s + 2.50·24-s + 8.63·25-s + 1.51·26-s − 0.742·27-s − 2.32·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.44·3-s + 0.5·4-s − 1.65·5-s + 1.02·6-s − 0.878·7-s − 0.353·8-s + 1.09·9-s + 1.16·10-s + 0.366·11-s − 0.724·12-s − 0.419·13-s + 0.621·14-s + 2.39·15-s + 0.250·16-s + 1.06·17-s − 0.776·18-s + 0.422·19-s − 0.825·20-s + 1.27·21-s − 0.258·22-s + 0.706·23-s + 0.512·24-s + 1.72·25-s + 0.296·26-s − 0.142·27-s − 0.439·28-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: \(0\)
Selberg data: \((2,\ 8002,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3254148487\)
\(L(\frac12)\) \(\approx\) \(0.3254148487\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.50T + 3T^{2} \)
5 \( 1 + 3.69T + 5T^{2} \)
7 \( 1 + 2.32T + 7T^{2} \)
11 \( 1 - 1.21T + 11T^{2} \)
13 \( 1 + 1.51T + 13T^{2} \)
17 \( 1 - 4.40T + 17T^{2} \)
19 \( 1 - 1.84T + 19T^{2} \)
23 \( 1 - 3.38T + 23T^{2} \)
29 \( 1 + 5.83T + 29T^{2} \)
31 \( 1 + 3.57T + 31T^{2} \)
37 \( 1 - 1.70T + 37T^{2} \)
41 \( 1 - 2.65T + 41T^{2} \)
43 \( 1 - 7.86T + 43T^{2} \)
47 \( 1 - 5.16T + 47T^{2} \)
53 \( 1 - 5.38T + 53T^{2} \)
59 \( 1 + 2.12T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 13.4T + 73T^{2} \)
79 \( 1 + 7.00T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + 9.16T + 89T^{2} \)
97 \( 1 + 2.69T + 97T^{2} \)
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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

−7.60601965371186328532669606773, −7.19644272521692624666185492399, −6.66092986739216333027554637856, −5.75342586593567968421340296274, −5.26199759291156437873913645973, −4.20998812623796742229298514691, −3.62071960862476914031453996684, −2.76144310457195681303054521897, −1.13596430382298707718110515390, −0.40934557332079275305356294915, 0.40934557332079275305356294915, 1.13596430382298707718110515390, 2.76144310457195681303054521897, 3.62071960862476914031453996684, 4.20998812623796742229298514691, 5.26199759291156437873913645973, 5.75342586593567968421340296274, 6.66092986739216333027554637856, 7.19644272521692624666185492399, 7.60601965371186328532669606773

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