Properties

Label 2-8020-1.1-c1-0-124
Degree $2$
Conductor $8020$
Sign $-1$
Analytic cond. $64.0400$
Root an. cond. $8.00250$
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  + 1.72·3-s + 5-s − 0.760·7-s − 0.0384·9-s − 0.652·11-s + 3.86·13-s + 1.72·15-s − 4.93·17-s − 1.98·19-s − 1.30·21-s + 1.36·23-s + 25-s − 5.22·27-s − 10.3·29-s − 3.31·31-s − 1.12·33-s − 0.760·35-s + 9.99·37-s + 6.64·39-s − 9.93·41-s − 5.32·43-s − 0.0384·45-s + 10.0·47-s − 6.42·49-s − 8.49·51-s + 14.4·53-s − 0.652·55-s + ⋯
L(s)  = 1  + 0.993·3-s + 0.447·5-s − 0.287·7-s − 0.0128·9-s − 0.196·11-s + 1.07·13-s + 0.444·15-s − 1.19·17-s − 0.455·19-s − 0.285·21-s + 0.284·23-s + 0.200·25-s − 1.00·27-s − 1.91·29-s − 0.595·31-s − 0.195·33-s − 0.128·35-s + 1.64·37-s + 1.06·39-s − 1.55·41-s − 0.811·43-s − 0.00572·45-s + 1.46·47-s − 0.917·49-s − 1.18·51-s + 1.97·53-s − 0.0880·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8020\)    =    \(2^{2} \cdot 5 \cdot 401\)
Sign: $-1$
Analytic conductor: \(64.0400\)
Root analytic conductor: \(8.00250\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8020,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
401 \( 1 + T \)
good3 \( 1 - 1.72T + 3T^{2} \)
7 \( 1 + 0.760T + 7T^{2} \)
11 \( 1 + 0.652T + 11T^{2} \)
13 \( 1 - 3.86T + 13T^{2} \)
17 \( 1 + 4.93T + 17T^{2} \)
19 \( 1 + 1.98T + 19T^{2} \)
23 \( 1 - 1.36T + 23T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 + 3.31T + 31T^{2} \)
37 \( 1 - 9.99T + 37T^{2} \)
41 \( 1 + 9.93T + 41T^{2} \)
43 \( 1 + 5.32T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 14.4T + 53T^{2} \)
59 \( 1 + 9.17T + 59T^{2} \)
61 \( 1 + 7.13T + 61T^{2} \)
67 \( 1 - 7.04T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 - 8.03T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 + 7.74T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−7.52469177736066391410131278112, −6.88925432869882049098231346685, −6.02579805348039558624226027005, −5.59828482979714429471910319353, −4.47225978001416791529452953568, −3.77942550061582804423131806391, −3.06295197803423268051780878423, −2.27544557810200926210440178317, −1.56504295349859344631299581215, 0, 1.56504295349859344631299581215, 2.27544557810200926210440178317, 3.06295197803423268051780878423, 3.77942550061582804423131806391, 4.47225978001416791529452953568, 5.59828482979714429471910319353, 6.02579805348039558624226027005, 6.88925432869882049098231346685, 7.52469177736066391410131278112

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