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Properties

Label 2-2020-2020.1183-c0-0-0
Degree $2$
Conductor $2020$
Sign $0.561 + 0.827i$
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  + (−0.876 − 0.481i)2-s + (0.535 + 0.844i)4-s + (−0.770 − 0.637i)5-s + (−0.0627 − 0.998i)8-s + (−0.968 − 0.248i)9-s + (0.368 + 0.929i)10-s + (0.689 + 1.01i)13-s + (−0.425 + 0.904i)16-s + (1.76 + 0.278i)17-s + (0.728 + 0.684i)18-s + (0.125 − 0.992i)20-s + (0.187 + 0.982i)25-s + (−0.115 − 1.22i)26-s + (−0.258 + 0.175i)29-s + (0.809 − 0.587i)32-s + ⋯
L(s)  = 1  + (−0.876 − 0.481i)2-s + (0.535 + 0.844i)4-s + (−0.770 − 0.637i)5-s + (−0.0627 − 0.998i)8-s + (−0.968 − 0.248i)9-s + (0.368 + 0.929i)10-s + (0.689 + 1.01i)13-s + (−0.425 + 0.904i)16-s + (1.76 + 0.278i)17-s + (0.728 + 0.684i)18-s + (0.125 − 0.992i)20-s + (0.187 + 0.982i)25-s + (−0.115 − 1.22i)26-s + (−0.258 + 0.175i)29-s + (0.809 − 0.587i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $0.561 + 0.827i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ 0.561 + 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6410309194\)
\(L(\frac12)\) \(\approx\) \(0.6410309194\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.876 + 0.481i)T \)
5 \( 1 + (0.770 + 0.637i)T \)
101 \( 1 + (0.844 - 0.535i)T \)
good3 \( 1 + (0.968 + 0.248i)T^{2} \)
7 \( 1 + (0.929 - 0.368i)T^{2} \)
11 \( 1 + (0.481 - 0.876i)T^{2} \)
13 \( 1 + (-0.689 - 1.01i)T + (-0.368 + 0.929i)T^{2} \)
17 \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \)
19 \( 1 + (-0.637 + 0.770i)T^{2} \)
23 \( 1 + (0.998 + 0.0627i)T^{2} \)
29 \( 1 + (0.258 - 0.175i)T + (0.368 - 0.929i)T^{2} \)
31 \( 1 + (0.929 - 0.368i)T^{2} \)
37 \( 1 + (-1.19 + 1.54i)T + (-0.248 - 0.968i)T^{2} \)
41 \( 1 + (0.294 + 1.85i)T + (-0.951 + 0.309i)T^{2} \)
43 \( 1 + (0.982 - 0.187i)T^{2} \)
47 \( 1 + (0.982 + 0.187i)T^{2} \)
53 \( 1 + (-1.05 + 1.65i)T + (-0.425 - 0.904i)T^{2} \)
59 \( 1 + (0.770 - 0.637i)T^{2} \)
61 \( 1 + (-0.288 + 1.29i)T + (-0.904 - 0.425i)T^{2} \)
67 \( 1 + (-0.968 + 0.248i)T^{2} \)
71 \( 1 + (-0.968 - 0.248i)T^{2} \)
73 \( 1 + (-0.872 - 0.929i)T + (-0.0627 + 0.998i)T^{2} \)
79 \( 1 + (0.0627 + 0.998i)T^{2} \)
83 \( 1 + (0.0627 + 0.998i)T^{2} \)
89 \( 1 + (-1.61 - 0.583i)T + (0.770 + 0.637i)T^{2} \)
97 \( 1 + (-1.61 - 0.360i)T + (0.904 + 0.425i)T^{2} \)
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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

−9.145609777609649070729602880942, −8.511902162870248150833299211965, −7.924692583905464569602111487311, −7.16544604217018594129899886343, −6.14700492512697739788306046434, −5.22862548838894159649078283411, −3.84070660388702728547442070889, −3.49393963603553792620768062015, −2.09459275156866634666094555605, −0.818735083025646438845100541830, 1.02206710113249041090231287051, 2.75584211318847732461777484288, 3.32942664122949759569150178453, 4.82891055500127015743984742230, 5.79344465531310779555748142485, 6.30105237846596595612062705345, 7.41494061854523023385691669294, 8.014453959358755510984993480202, 8.329610303478969561767793895190, 9.423269635837987383377559769710

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