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Properties

Label 2-2020-2020.1063-c0-0-0
Degree $2$
Conductor $2020$
Sign $0.881 - 0.471i$
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  + (−0.998 + 0.0627i)2-s + (0.992 − 0.125i)4-s + (−0.368 − 0.929i)5-s + (−0.982 + 0.187i)8-s + (0.728 + 0.684i)9-s + (0.425 + 0.904i)10-s + (0.400 + 1.79i)13-s + (0.968 − 0.248i)16-s + (0.142 − 0.278i)17-s + (−0.770 − 0.637i)18-s + (−0.481 − 0.876i)20-s + (−0.728 + 0.684i)25-s + (−0.512 − 1.76i)26-s + (0.198 + 0.886i)29-s + (−0.951 + 0.309i)32-s + ⋯
L(s)  = 1  + (−0.998 + 0.0627i)2-s + (0.992 − 0.125i)4-s + (−0.368 − 0.929i)5-s + (−0.982 + 0.187i)8-s + (0.728 + 0.684i)9-s + (0.425 + 0.904i)10-s + (0.400 + 1.79i)13-s + (0.968 − 0.248i)16-s + (0.142 − 0.278i)17-s + (−0.770 − 0.637i)18-s + (−0.481 − 0.876i)20-s + (−0.728 + 0.684i)25-s + (−0.512 − 1.76i)26-s + (0.198 + 0.886i)29-s + (−0.951 + 0.309i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1063, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ 0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7315819803\)
\(L(\frac12)\) \(\approx\) \(0.7315819803\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.998 - 0.0627i)T \)
5 \( 1 + (0.368 + 0.929i)T \)
101 \( 1 + (-0.125 - 0.992i)T \)
good3 \( 1 + (-0.728 - 0.684i)T^{2} \)
7 \( 1 + (-0.425 + 0.904i)T^{2} \)
11 \( 1 + (-0.998 + 0.0627i)T^{2} \)
13 \( 1 + (-0.400 - 1.79i)T + (-0.904 + 0.425i)T^{2} \)
17 \( 1 + (-0.142 + 0.278i)T + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (0.876 + 0.481i)T^{2} \)
23 \( 1 + (-0.982 - 0.187i)T^{2} \)
29 \( 1 + (-0.198 - 0.886i)T + (-0.904 + 0.425i)T^{2} \)
31 \( 1 + (0.425 - 0.904i)T^{2} \)
37 \( 1 + (0.627 - 1.45i)T + (-0.684 - 0.728i)T^{2} \)
41 \( 1 + (-0.462 - 0.907i)T + (-0.587 + 0.809i)T^{2} \)
43 \( 1 + (-0.844 + 0.535i)T^{2} \)
47 \( 1 + (-0.844 - 0.535i)T^{2} \)
53 \( 1 + (-0.211 + 1.67i)T + (-0.968 - 0.248i)T^{2} \)
59 \( 1 + (-0.481 - 0.876i)T^{2} \)
61 \( 1 + (-1.01 + 1.30i)T + (-0.248 - 0.968i)T^{2} \)
67 \( 1 + (0.728 - 0.684i)T^{2} \)
71 \( 1 + (-0.728 - 0.684i)T^{2} \)
73 \( 1 + (-1.11 - 1.35i)T + (-0.187 + 0.982i)T^{2} \)
79 \( 1 + (-0.187 - 0.982i)T^{2} \)
83 \( 1 + (0.187 + 0.982i)T^{2} \)
89 \( 1 + (-0.0319 - 0.0540i)T + (-0.481 + 0.876i)T^{2} \)
97 \( 1 + (0.267 - 0.344i)T + (-0.248 - 0.968i)T^{2} \)
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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

−9.447126751155224700595778840017, −8.486277690652541650480809231723, −8.160534393796013996064033839327, −7.03353692828894816354555829305, −6.68161321720863095597524026261, −5.36922870883994319049096589850, −4.59818974908821716976949817781, −3.59243591852342609640471515647, −2.07725890920281055137174872164, −1.28892339160645948697759309871, 0.834962832165782362881465201351, 2.35122025482509592285328336020, 3.28286431643945396708436734020, 4.01263079473682067746635559971, 5.69803442691086896050895186522, 6.23487849576197086357010636419, 7.27875442021730502557561998788, 7.60720911258315768579752555355, 8.470359804201097659020661219667, 9.312402671350824556877129613465

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