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Properties

Label 2-2020-2020.1259-c0-0-2
Degree $2$
Conductor $2020$
Sign $-0.574 + 0.818i$
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  + (−0.968 − 0.248i)2-s + (0.876 + 0.481i)4-s + (−0.0627 − 0.998i)5-s + (−0.728 − 0.684i)8-s + (−0.992 − 0.125i)9-s + (−0.187 + 0.982i)10-s + (1.53 − 1.27i)13-s + (0.535 + 0.844i)16-s + (−1.11 + 0.363i)17-s + (0.929 + 0.368i)18-s + (0.425 − 0.904i)20-s + (−0.992 + 0.125i)25-s + (−1.80 + 0.849i)26-s + (1.46 − 1.21i)29-s + (−0.309 − 0.951i)32-s + ⋯
L(s)  = 1  + (−0.968 − 0.248i)2-s + (0.876 + 0.481i)4-s + (−0.0627 − 0.998i)5-s + (−0.728 − 0.684i)8-s + (−0.992 − 0.125i)9-s + (−0.187 + 0.982i)10-s + (1.53 − 1.27i)13-s + (0.535 + 0.844i)16-s + (−1.11 + 0.363i)17-s + (0.929 + 0.368i)18-s + (0.425 − 0.904i)20-s + (−0.992 + 0.125i)25-s + (−1.80 + 0.849i)26-s + (1.46 − 1.21i)29-s + (−0.309 − 0.951i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $-0.574 + 0.818i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ -0.574 + 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5666040372\)
\(L(\frac12)\) \(\approx\) \(0.5666040372\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.968 + 0.248i)T \)
5 \( 1 + (0.0627 + 0.998i)T \)
101 \( 1 + (0.876 + 0.481i)T \)
good3 \( 1 + (0.992 + 0.125i)T^{2} \)
7 \( 1 + (0.187 + 0.982i)T^{2} \)
11 \( 1 + (0.968 + 0.248i)T^{2} \)
13 \( 1 + (-1.53 + 1.27i)T + (0.187 - 0.982i)T^{2} \)
17 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.425 + 0.904i)T^{2} \)
23 \( 1 + (0.728 - 0.684i)T^{2} \)
29 \( 1 + (-1.46 + 1.21i)T + (0.187 - 0.982i)T^{2} \)
31 \( 1 + (0.187 + 0.982i)T^{2} \)
37 \( 1 + (0.961 - 0.0604i)T + (0.992 - 0.125i)T^{2} \)
41 \( 1 + (1.60 + 0.521i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.637 + 0.770i)T^{2} \)
47 \( 1 + (-0.637 - 0.770i)T^{2} \)
53 \( 1 + (1.11 - 0.614i)T + (0.535 - 0.844i)T^{2} \)
59 \( 1 + (-0.425 + 0.904i)T^{2} \)
61 \( 1 + (-0.659 + 1.19i)T + (-0.535 - 0.844i)T^{2} \)
67 \( 1 + (0.992 - 0.125i)T^{2} \)
71 \( 1 + (0.992 + 0.125i)T^{2} \)
73 \( 1 + (-0.791 + 0.313i)T + (0.728 - 0.684i)T^{2} \)
79 \( 1 + (-0.728 - 0.684i)T^{2} \)
83 \( 1 + (-0.728 - 0.684i)T^{2} \)
89 \( 1 + (-0.211 - 0.134i)T + (0.425 + 0.904i)T^{2} \)
97 \( 1 + (0.742 - 1.35i)T + (-0.535 - 0.844i)T^{2} \)
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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

−8.845802346910582251064072111499, −8.279829573175036284938338521167, −8.151190278051952779593950681656, −6.67965036901653682319404556802, −6.05076721551091064249275847209, −5.17926084656166360945029108170, −3.89509826609023880947004886916, −3.05017303667559357290461633016, −1.81704348229729772435370266663, −0.55058441269973918935800658821, 1.62823513087312871663602490899, 2.69281194776666224036641146537, 3.55528724298466037213522300152, 4.92903771753571213578357740248, 6.11202626254601045158274780412, 6.57654090174249810442328300521, 7.13306854637086014083939101090, 8.381363434054815407984440324345, 8.635955818674763766559491048269, 9.457489896203187505663551023113

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