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Properties

Label 2-2020-2020.1419-c0-0-1
Degree $2$
Conductor $2020$
Sign $0.778 + 0.628i$
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  + (0.728 + 0.684i)2-s + (0.371 − 1.94i)3-s + (0.0627 + 0.998i)4-s + (0.728 − 0.684i)5-s + (1.60 − 1.16i)6-s + (1.27 + 0.702i)7-s + (−0.637 + 0.770i)8-s + (−2.73 − 1.08i)9-s + 10-s + (1.96 + 0.248i)12-s + (0.450 + 1.38i)14-s + (−1.06 − 1.67i)15-s + (−0.992 + 0.125i)16-s + (−1.25 − 2.65i)18-s + (0.728 + 0.684i)20-s + (1.84 − 2.22i)21-s + ⋯
L(s)  = 1  + (0.728 + 0.684i)2-s + (0.371 − 1.94i)3-s + (0.0627 + 0.998i)4-s + (0.728 − 0.684i)5-s + (1.60 − 1.16i)6-s + (1.27 + 0.702i)7-s + (−0.637 + 0.770i)8-s + (−2.73 − 1.08i)9-s + 10-s + (1.96 + 0.248i)12-s + (0.450 + 1.38i)14-s + (−1.06 − 1.67i)15-s + (−0.992 + 0.125i)16-s + (−1.25 − 2.65i)18-s + (0.728 + 0.684i)20-s + (1.84 − 2.22i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $0.778 + 0.628i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ 0.778 + 0.628i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.235536976\)
\(L(\frac12)\) \(\approx\) \(2.235536976\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.728 - 0.684i)T \)
5 \( 1 + (-0.728 + 0.684i)T \)
101 \( 1 + (0.637 - 0.770i)T \)
good3 \( 1 + (-0.371 + 1.94i)T + (-0.929 - 0.368i)T^{2} \)
7 \( 1 + (-1.27 - 0.702i)T + (0.535 + 0.844i)T^{2} \)
11 \( 1 + (-0.728 - 0.684i)T^{2} \)
13 \( 1 + (-0.535 + 0.844i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.968 - 0.248i)T^{2} \)
23 \( 1 + (0.456 - 0.969i)T + (-0.637 - 0.770i)T^{2} \)
29 \( 1 + (-1.27 + 0.702i)T + (0.535 - 0.844i)T^{2} \)
31 \( 1 + (-0.535 - 0.844i)T^{2} \)
37 \( 1 + (0.929 - 0.368i)T^{2} \)
41 \( 1 + (1.41 - 1.03i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (0.362 + 0.0931i)T + (0.876 + 0.481i)T^{2} \)
47 \( 1 + (1.92 - 0.493i)T + (0.876 - 0.481i)T^{2} \)
53 \( 1 + (0.992 + 0.125i)T^{2} \)
59 \( 1 + (-0.968 + 0.248i)T^{2} \)
61 \( 1 + (0.116 + 1.85i)T + (-0.992 + 0.125i)T^{2} \)
67 \( 1 + (-0.159 - 0.836i)T + (-0.929 + 0.368i)T^{2} \)
71 \( 1 + (0.929 + 0.368i)T^{2} \)
73 \( 1 + (0.637 + 0.770i)T^{2} \)
79 \( 1 + (0.637 - 0.770i)T^{2} \)
83 \( 1 + (-0.159 - 0.339i)T + (-0.637 + 0.770i)T^{2} \)
89 \( 1 + (0.613 + 0.0774i)T + (0.968 + 0.248i)T^{2} \)
97 \( 1 + (0.992 - 0.125i)T^{2} \)
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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

−8.650977771173685512784971659198, −8.214803712361322316549233480773, −7.83725930296617251291667915161, −6.75249929369322922146232437307, −6.18296458293624171351995231496, −5.43570708998103571697311609130, −4.78330465583897393505535369212, −3.16944514567235848706308463219, −2.17113220640618442865180770274, −1.51399349904571371710231539578, 1.89994562646484229093215427191, 2.91235846846905223749465329643, 3.66163466188457581145799768916, 4.58683919486728975865263410322, 4.97418508945653255392449123775, 5.81483365689880345463153159979, 6.84480529591722461526233949528, 8.231728722600434054720736315856, 8.868144115665294768010484718227, 9.856431590039571800503445470510

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