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Properties

Label 2-2020-2020.1059-c0-0-3
Degree $2$
Conductor $2020$
Sign $-0.687 + 0.725i$
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  + (0.535 − 0.844i)2-s + (−0.425 − 0.904i)4-s + (0.187 − 0.982i)5-s + (−0.992 − 0.125i)8-s + (0.876 − 0.481i)9-s + (−0.728 − 0.684i)10-s + (0.183 + 0.462i)13-s + (−0.637 + 0.770i)16-s + (1.11 − 0.363i)17-s + (0.0627 − 0.998i)18-s + (−0.968 + 0.248i)20-s + (−0.929 − 0.368i)25-s + (0.488 + 0.0931i)26-s + (−0.700 − 1.76i)29-s + (0.309 + 0.951i)32-s + ⋯
L(s)  = 1  + (0.535 − 0.844i)2-s + (−0.425 − 0.904i)4-s + (0.187 − 0.982i)5-s + (−0.992 − 0.125i)8-s + (0.876 − 0.481i)9-s + (−0.728 − 0.684i)10-s + (0.183 + 0.462i)13-s + (−0.637 + 0.770i)16-s + (1.11 − 0.363i)17-s + (0.0627 − 0.998i)18-s + (−0.968 + 0.248i)20-s + (−0.929 − 0.368i)25-s + (0.488 + 0.0931i)26-s + (−0.700 − 1.76i)29-s + (0.309 + 0.951i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $-0.687 + 0.725i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ -0.687 + 0.725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.550211789\)
\(L(\frac12)\) \(\approx\) \(1.550211789\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.535 + 0.844i)T \)
5 \( 1 + (-0.187 + 0.982i)T \)
101 \( 1 + (-0.425 - 0.904i)T \)
good3 \( 1 + (-0.876 + 0.481i)T^{2} \)
7 \( 1 + (-0.728 - 0.684i)T^{2} \)
11 \( 1 + (0.535 - 0.844i)T^{2} \)
13 \( 1 + (-0.183 - 0.462i)T + (-0.728 + 0.684i)T^{2} \)
17 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.187 - 0.982i)T^{2} \)
23 \( 1 + (-0.992 + 0.125i)T^{2} \)
29 \( 1 + (0.700 + 1.76i)T + (-0.728 + 0.684i)T^{2} \)
31 \( 1 + (-0.728 - 0.684i)T^{2} \)
37 \( 1 + (0.450 - 1.75i)T + (-0.876 - 0.481i)T^{2} \)
41 \( 1 + (1.46 + 0.476i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.929 - 0.368i)T^{2} \)
47 \( 1 + (-0.929 + 0.368i)T^{2} \)
53 \( 1 + (0.791 - 1.68i)T + (-0.637 - 0.770i)T^{2} \)
59 \( 1 + (-0.187 - 0.982i)T^{2} \)
61 \( 1 + (-0.226 + 0.106i)T + (0.637 - 0.770i)T^{2} \)
67 \( 1 + (-0.876 - 0.481i)T^{2} \)
71 \( 1 + (-0.876 + 0.481i)T^{2} \)
73 \( 1 + (-0.0235 - 0.374i)T + (-0.992 + 0.125i)T^{2} \)
79 \( 1 + (0.992 + 0.125i)T^{2} \)
83 \( 1 + (0.992 + 0.125i)T^{2} \)
89 \( 1 + (-0.742 + 0.614i)T + (0.187 - 0.982i)T^{2} \)
97 \( 1 + (-0.666 + 0.313i)T + (0.637 - 0.770i)T^{2} \)
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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

−9.337659643773199396569080924620, −8.498892507585766035257658688176, −7.53381455993063115436730660292, −6.41606965106884940172549788100, −5.68646640677920745618786718242, −4.78847275602519024995426384211, −4.16165682035349525652626337589, −3.26694245998527315712966562810, −1.91958624505305150891481313672, −1.03219329527045227367982916430, 1.92665923300601073998304202304, 3.28291186385237155080204072113, 3.76209377638089366316912221827, 5.04896869288950517882495024248, 5.61306756109799506559862694339, 6.58978302363353292585138930554, 7.21896506709682918092641561523, 7.76383060817931138664588451812, 8.627963091574457465035770982343, 9.626586571198727245048508005372

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