Properties

Label 2-8020-1.1-c1-0-58
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.795·3-s + 5-s + 2.08·7-s − 2.36·9-s + 3.33·11-s + 3.89·13-s + 0.795·15-s − 6.62·17-s + 1.82·19-s + 1.65·21-s + 5.82·23-s + 25-s − 4.27·27-s + 2.07·29-s + 5.43·31-s + 2.65·33-s + 2.08·35-s + 0.0603·37-s + 3.09·39-s + 3.93·41-s + 3.95·43-s − 2.36·45-s + 1.86·47-s − 2.65·49-s − 5.26·51-s − 5.15·53-s + 3.33·55-s + ⋯
L(s)  = 1  + 0.459·3-s + 0.447·5-s + 0.787·7-s − 0.788·9-s + 1.00·11-s + 1.07·13-s + 0.205·15-s − 1.60·17-s + 0.417·19-s + 0.361·21-s + 1.21·23-s + 0.200·25-s − 0.821·27-s + 0.385·29-s + 0.976·31-s + 0.462·33-s + 0.352·35-s + 0.00992·37-s + 0.496·39-s + 0.615·41-s + 0.602·43-s − 0.352·45-s + 0.271·47-s − 0.379·49-s − 0.737·51-s − 0.708·53-s + 0.449·55-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: \(0\)
Selberg data: \((2,\ 8020,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.238005001\)
\(L(\frac12)\) \(\approx\) \(3.238005001\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.795T + 3T^{2} \)
7 \( 1 - 2.08T + 7T^{2} \)
11 \( 1 - 3.33T + 11T^{2} \)
13 \( 1 - 3.89T + 13T^{2} \)
17 \( 1 + 6.62T + 17T^{2} \)
19 \( 1 - 1.82T + 19T^{2} \)
23 \( 1 - 5.82T + 23T^{2} \)
29 \( 1 - 2.07T + 29T^{2} \)
31 \( 1 - 5.43T + 31T^{2} \)
37 \( 1 - 0.0603T + 37T^{2} \)
41 \( 1 - 3.93T + 41T^{2} \)
43 \( 1 - 3.95T + 43T^{2} \)
47 \( 1 - 1.86T + 47T^{2} \)
53 \( 1 + 5.15T + 53T^{2} \)
59 \( 1 + 4.27T + 59T^{2} \)
61 \( 1 + 6.28T + 61T^{2} \)
67 \( 1 - 1.22T + 67T^{2} \)
71 \( 1 - 3.52T + 71T^{2} \)
73 \( 1 - 4.30T + 73T^{2} \)
79 \( 1 - 3.14T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + 8.10T + 89T^{2} \)
97 \( 1 + 8.32T + 97T^{2} \)
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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

−8.076604812654119410825014629781, −7.07435495331878533676985271644, −6.40452635271566009474821406427, −5.89554820897161990392299455733, −4.92421524691759978187104930452, −4.34457382305491904693732066864, −3.43051645240995144185440793664, −2.65658903082699757330158330774, −1.79161840389080714118537948875, −0.917194649431653710372475840887, 0.917194649431653710372475840887, 1.79161840389080714118537948875, 2.65658903082699757330158330774, 3.43051645240995144185440793664, 4.34457382305491904693732066864, 4.92421524691759978187104930452, 5.89554820897161990392299455733, 6.40452635271566009474821406427, 7.07435495331878533676985271644, 8.076604812654119410825014629781

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