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Properties

Label 2-2020-2020.143-c0-0-0
Degree $2$
Conductor $2020$
Sign $-0.845 + 0.534i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.125 − 0.992i)2-s + (−0.968 − 0.248i)4-s + (−0.684 + 0.728i)5-s + (−0.368 + 0.929i)8-s + (0.0627 − 0.998i)9-s + (0.637 + 0.770i)10-s + (0.0212 + 0.0591i)13-s + (0.876 + 0.481i)16-s + (−0.278 − 1.76i)17-s + (−0.982 − 0.187i)18-s + (0.844 − 0.535i)20-s + (−0.0627 − 0.998i)25-s + (0.0613 − 0.0137i)26-s + (−0.105 − 0.294i)29-s + (0.587 − 0.809i)32-s + ⋯
L(s)  = 1  + (0.125 − 0.992i)2-s + (−0.968 − 0.248i)4-s + (−0.684 + 0.728i)5-s + (−0.368 + 0.929i)8-s + (0.0627 − 0.998i)9-s + (0.637 + 0.770i)10-s + (0.0212 + 0.0591i)13-s + (0.876 + 0.481i)16-s + (−0.278 − 1.76i)17-s + (−0.982 − 0.187i)18-s + (0.844 − 0.535i)20-s + (−0.0627 − 0.998i)25-s + (0.0613 − 0.0137i)26-s + (−0.105 − 0.294i)29-s + (0.587 − 0.809i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $-0.845 + 0.534i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ -0.845 + 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7407989662\)
\(L(\frac12)\) \(\approx\) \(0.7407989662\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.125 + 0.992i)T \)
5 \( 1 + (0.684 - 0.728i)T \)
101 \( 1 + (-0.248 + 0.968i)T \)
good3 \( 1 + (-0.0627 + 0.998i)T^{2} \)
7 \( 1 + (-0.637 + 0.770i)T^{2} \)
11 \( 1 + (0.125 - 0.992i)T^{2} \)
13 \( 1 + (-0.0212 - 0.0591i)T + (-0.770 + 0.637i)T^{2} \)
17 \( 1 + (0.278 + 1.76i)T + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.535 - 0.844i)T^{2} \)
23 \( 1 + (-0.368 - 0.929i)T^{2} \)
29 \( 1 + (0.105 + 0.294i)T + (-0.770 + 0.637i)T^{2} \)
31 \( 1 + (0.637 - 0.770i)T^{2} \)
37 \( 1 + (1.01 - 0.0319i)T + (0.998 - 0.0627i)T^{2} \)
41 \( 1 + (-0.300 + 1.89i)T + (-0.951 - 0.309i)T^{2} \)
43 \( 1 + (0.904 - 0.425i)T^{2} \)
47 \( 1 + (0.904 + 0.425i)T^{2} \)
53 \( 1 + (0.450 + 1.75i)T + (-0.876 + 0.481i)T^{2} \)
59 \( 1 + (0.844 - 0.535i)T^{2} \)
61 \( 1 + (1.57 - 0.934i)T + (0.481 - 0.876i)T^{2} \)
67 \( 1 + (0.0627 + 0.998i)T^{2} \)
71 \( 1 + (-0.0627 + 0.998i)T^{2} \)
73 \( 1 + (-0.200 - 1.05i)T + (-0.929 + 0.368i)T^{2} \)
79 \( 1 + (-0.929 - 0.368i)T^{2} \)
83 \( 1 + (0.929 + 0.368i)T^{2} \)
89 \( 1 + (-1.44 - 0.418i)T + (0.844 + 0.535i)T^{2} \)
97 \( 1 + (0.583 - 0.344i)T + (0.481 - 0.876i)T^{2} \)
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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

−9.158186404569149013541378345406, −8.496049644528909626733725887952, −7.38711268212104452796226939306, −6.78340129355732966876268341107, −5.68883085865601040733773609545, −4.71621847731822502102344289434, −3.80427000803610661217156986140, −3.17579339119249739179135035015, −2.21742267403296490692318594906, −0.53505037043960758402297247010, 1.53626941048480138839795127748, 3.26494790630643230448258915538, 4.28325824216071065648052605447, 4.78101323679061549319640546328, 5.71796089589798243596573976998, 6.45807143277481302330363398265, 7.60576669017267282834541002403, 7.920977776341903807511627671405, 8.672273501754602157272005784590, 9.289987462095288405324761079293

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