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Properties

Label 1-2020-2020.979-r1-0-0
Degree $1$
Conductor $2020$
Sign $0.126 + 0.991i$
Analytic cond. $217.079$
Root an. cond. $217.079$
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  + (−0.968 + 0.248i)3-s + (0.929 + 0.368i)7-s + (0.876 − 0.481i)9-s + (0.876 − 0.481i)11-s + (0.929 − 0.368i)13-s + (−0.309 − 0.951i)17-s + (0.637 + 0.770i)19-s + (−0.992 − 0.125i)21-s + (0.0627 + 0.998i)23-s + (−0.728 + 0.684i)27-s + (0.929 − 0.368i)29-s + (0.929 + 0.368i)31-s + (−0.728 + 0.684i)33-s + (−0.968 − 0.248i)37-s + (−0.809 + 0.587i)39-s + ⋯
L(s)  = 1  + (−0.968 + 0.248i)3-s + (0.929 + 0.368i)7-s + (0.876 − 0.481i)9-s + (0.876 − 0.481i)11-s + (0.929 − 0.368i)13-s + (−0.309 − 0.951i)17-s + (0.637 + 0.770i)19-s + (−0.992 − 0.125i)21-s + (0.0627 + 0.998i)23-s + (−0.728 + 0.684i)27-s + (0.929 − 0.368i)29-s + (0.929 + 0.368i)31-s + (−0.728 + 0.684i)33-s + (−0.968 − 0.248i)37-s + (−0.809 + 0.587i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $0.126 + 0.991i$
Analytic conductor: \(217.079\)
Root analytic conductor: \(217.079\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2020,\ (1:\ ),\ 0.126 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.469250331 + 1.293485317i\)
\(L(\frac12)\) \(\approx\) \(1.469250331 + 1.293485317i\)
\(L(1)\) \(\approx\) \(1.008713438 + 0.1842851528i\)
\(L(1)\) \(\approx\) \(1.008713438 + 0.1842851528i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
101 \( 1 \)
good3 \( 1 + (-0.968 + 0.248i)T \)
7 \( 1 + (0.929 + 0.368i)T \)
11 \( 1 + (0.876 - 0.481i)T \)
13 \( 1 + (0.929 - 0.368i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.637 + 0.770i)T \)
23 \( 1 + (0.0627 + 0.998i)T \)
29 \( 1 + (0.929 - 0.368i)T \)
31 \( 1 + (0.929 + 0.368i)T \)
37 \( 1 + (-0.968 - 0.248i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + (-0.187 + 0.982i)T \)
47 \( 1 + (-0.187 - 0.982i)T \)
53 \( 1 + (-0.425 + 0.904i)T \)
59 \( 1 + (-0.637 + 0.770i)T \)
61 \( 1 + (0.425 + 0.904i)T \)
67 \( 1 + (-0.968 - 0.248i)T \)
71 \( 1 + (-0.968 + 0.248i)T \)
73 \( 1 + (0.0627 + 0.998i)T \)
79 \( 1 + (-0.0627 + 0.998i)T \)
83 \( 1 + (-0.0627 + 0.998i)T \)
89 \( 1 + (0.637 + 0.770i)T \)
97 \( 1 + (0.425 + 0.904i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.43586492781396942503489804716, −18.7956575273361384888547948857, −17.86358193361499258505874072567, −17.468795729014839364948717127134, −16.914755236735116049320837765438, −15.98226733096514698773252657729, −15.349436828391977927780988091585, −14.344126575200268606658194243688, −13.734486751718295157679655991977, −12.88302836076245543212862128736, −11.99206932720180685037184368801, −11.552785466125881594203926131239, −10.70680768811243473243561169491, −10.24726355648578369355959410199, −9.01185731334766095846391690162, −8.33695034772876005417581748299, −7.3408148052580178904204165798, −6.620128374048519217823596747680, −6.06589687205568727326069882008, −4.848308725053782320452376594487, −4.50449584504473542920057648380, −3.51218864112370063662987123315, −1.96397583083720185613135871288, −1.380417747625360760509259157317, −0.4534682781414826655284919666, 1.05263921578199612566165547544, 1.394269093121833301773294586605, 2.92673242483172521847426369322, 3.876576450886178754518509789841, 4.71288744637139940551810277143, 5.475278481949313320687484062097, 6.1151173401360174093277221105, 6.938659704452523597852449085938, 7.91784006473342445267940247824, 8.7206407798204348623108321550, 9.551117986510177577675180156174, 10.392930862196727279724528405, 11.21561158750704029124971311419, 11.75040306268635947178371854026, 12.15894141435954478321546561460, 13.44432127260287466524760667397, 13.96114777565952314003558280448, 14.94457131685700344141579713836, 15.701238800363272178231444153005, 16.23811000557573474352636551404, 17.0750777107010567646693909767, 17.83755774944400932016772228971, 18.17785594627729866840882649868, 19.024291126531318389501339882455, 19.98936335243001643130623523119

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