↑

Properties

Label 2-2020-2020.1019-c0-0-0
Degree $2$
Conductor $2020$
Sign $0.997 - 0.0683i$
Analytic cond. $1.00811$
Root an. cond. $1.00404$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.368 − 0.929i)2-s + (0.0967 + 0.0800i)3-s + (−0.728 + 0.684i)4-s + (0.929 + 0.368i)5-s + (0.0388 − 0.119i)6-s + (0.462 + 1.80i)7-s + (0.904 + 0.425i)8-s + (−0.184 − 0.966i)9-s − 1.00i·10-s + (−0.125 + 0.00788i)12-s + (1.50 − 1.09i)14-s + (0.0604 + 0.110i)15-s + (0.0627 − 0.998i)16-s + (−0.831 + 0.527i)18-s + (−0.929 + 0.368i)20-s + (−0.0994 + 0.211i)21-s + ⋯
L(s)  = 1  + (−0.368 − 0.929i)2-s + (0.0967 + 0.0800i)3-s + (−0.728 + 0.684i)4-s + (0.929 + 0.368i)5-s + (0.0388 − 0.119i)6-s + (0.462 + 1.80i)7-s + (0.904 + 0.425i)8-s + (−0.184 − 0.966i)9-s − 1.00i·10-s + (−0.125 + 0.00788i)12-s + (1.50 − 1.09i)14-s + (0.0604 + 0.110i)15-s + (0.0627 − 0.998i)16-s + (−0.831 + 0.527i)18-s + (−0.929 + 0.368i)20-s + (−0.0994 + 0.211i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $0.997 - 0.0683i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ 0.997 - 0.0683i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.125934403\)
\(L(\frac12)\) \(\approx\) \(1.125934403\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.368 + 0.929i)T \)
5 \( 1 + (-0.929 - 0.368i)T \)
101 \( 1 + (-0.425 + 0.904i)T \)
good3 \( 1 + (-0.0967 - 0.0800i)T + (0.187 + 0.982i)T^{2} \)
7 \( 1 + (-0.462 - 1.80i)T + (-0.876 + 0.481i)T^{2} \)
11 \( 1 + (-0.929 + 0.368i)T^{2} \)
13 \( 1 + (-0.876 - 0.481i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.992 - 0.125i)T^{2} \)
23 \( 1 + (0.516 - 0.813i)T + (-0.425 - 0.904i)T^{2} \)
29 \( 1 + (-0.183 + 0.713i)T + (-0.876 - 0.481i)T^{2} \)
31 \( 1 + (-0.876 + 0.481i)T^{2} \)
37 \( 1 + (0.187 - 0.982i)T^{2} \)
41 \( 1 + (-0.473 - 0.153i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (1.52 - 0.193i)T + (0.968 - 0.248i)T^{2} \)
47 \( 1 + (-1.98 - 0.250i)T + (0.968 + 0.248i)T^{2} \)
53 \( 1 + (0.0627 + 0.998i)T^{2} \)
59 \( 1 + (-0.992 - 0.125i)T^{2} \)
61 \( 1 + (-1.34 - 1.43i)T + (-0.0627 + 0.998i)T^{2} \)
67 \( 1 + (0.825 - 0.683i)T + (0.187 - 0.982i)T^{2} \)
71 \( 1 + (0.187 + 0.982i)T^{2} \)
73 \( 1 + (-0.425 - 0.904i)T^{2} \)
79 \( 1 + (0.425 - 0.904i)T^{2} \)
83 \( 1 + (-1.07 + 0.683i)T + (0.425 - 0.904i)T^{2} \)
89 \( 1 + (1.17 - 0.0738i)T + (0.992 - 0.125i)T^{2} \)
97 \( 1 + (-0.0627 + 0.998i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.457276741375519125356443322443, −8.781759967108513314403159009860, −8.233118950009217939377787304943, −7.04075412911325898118908035566, −5.87887043203430742374426193236, −5.51527532678169668789829838218, −4.30203749047520569568251037603, −3.10235163709075552224518648590, −2.47146843373040130257554384990, −1.56328230629353963407274287686, 1.01258832523043235304829885086, 2.09570325101814821265688993866, 3.85597254894406422595110584949, 4.75996044617240909284209080367, 5.26484621646908033767286510347, 6.36532368821677952728052211542, 7.02684848221986386451520001232, 7.79263272066635090257853788578, 8.367479886640098090310480976397, 9.195323543283610842022267482166

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