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Properties

Label 2-2020-2020.1019-c0-0-3
Degree $2$
Conductor $2020$
Sign $0.688 + 0.725i$
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  + (0.929 − 0.368i)2-s + (0.728 − 0.684i)4-s + (0.637 + 0.770i)5-s + (0.425 − 0.904i)8-s + (−0.187 − 0.982i)9-s + (0.876 + 0.481i)10-s + (0.383 − 1.49i)13-s + (0.0627 − 0.998i)16-s + (−1.11 + 0.363i)17-s + (−0.535 − 0.844i)18-s + (0.992 + 0.125i)20-s + (−0.187 + 0.982i)25-s + (−0.193 − 1.52i)26-s + (−0.473 + 1.84i)29-s + (−0.309 − 0.951i)32-s + ⋯
L(s)  = 1  + (0.929 − 0.368i)2-s + (0.728 − 0.684i)4-s + (0.637 + 0.770i)5-s + (0.425 − 0.904i)8-s + (−0.187 − 0.982i)9-s + (0.876 + 0.481i)10-s + (0.383 − 1.49i)13-s + (0.0627 − 0.998i)16-s + (−1.11 + 0.363i)17-s + (−0.535 − 0.844i)18-s + (0.992 + 0.125i)20-s + (−0.187 + 0.982i)25-s + (−0.193 − 1.52i)26-s + (−0.473 + 1.84i)29-s + (−0.309 − 0.951i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $0.688 + 0.725i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ 0.688 + 0.725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.218929385\)
\(L(\frac12)\) \(\approx\) \(2.218929385\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.929 + 0.368i)T \)
5 \( 1 + (-0.637 - 0.770i)T \)
101 \( 1 + (0.728 - 0.684i)T \)
good3 \( 1 + (0.187 + 0.982i)T^{2} \)
7 \( 1 + (-0.876 + 0.481i)T^{2} \)
11 \( 1 + (-0.929 + 0.368i)T^{2} \)
13 \( 1 + (-0.383 + 1.49i)T + (-0.876 - 0.481i)T^{2} \)
17 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.992 - 0.125i)T^{2} \)
23 \( 1 + (-0.425 - 0.904i)T^{2} \)
29 \( 1 + (0.473 - 1.84i)T + (-0.876 - 0.481i)T^{2} \)
31 \( 1 + (-0.876 + 0.481i)T^{2} \)
37 \( 1 + (1.05 - 0.872i)T + (0.187 - 0.982i)T^{2} \)
41 \( 1 + (-1.89 - 0.616i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.968 - 0.248i)T^{2} \)
47 \( 1 + (0.968 + 0.248i)T^{2} \)
53 \( 1 + (-1.41 - 1.32i)T + (0.0627 + 0.998i)T^{2} \)
59 \( 1 + (-0.992 - 0.125i)T^{2} \)
61 \( 1 + (1.23 + 1.31i)T + (-0.0627 + 0.998i)T^{2} \)
67 \( 1 + (0.187 - 0.982i)T^{2} \)
71 \( 1 + (0.187 + 0.982i)T^{2} \)
73 \( 1 + (1.06 - 1.67i)T + (-0.425 - 0.904i)T^{2} \)
79 \( 1 + (0.425 - 0.904i)T^{2} \)
83 \( 1 + (0.425 - 0.904i)T^{2} \)
89 \( 1 + (1.96 - 0.123i)T + (0.992 - 0.125i)T^{2} \)
97 \( 1 + (0.340 + 0.362i)T + (-0.0627 + 0.998i)T^{2} \)
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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

−9.397962969724480937285246573567, −8.583483539926867834998669414714, −7.30640948927594733376330303243, −6.70315828952101537688542148265, −5.90313667645113092146295478052, −5.42121208918213213097783293196, −4.17714169119637598742882795022, −3.26624497797655173666598901516, −2.68484587062121514153313110054, −1.36453393855869085268212271173, 1.91852019738582346967811984582, 2.45049166765005747526225587802, 4.15169165304621479814455809106, 4.44853732419436480548766555233, 5.50401547796543557458956091934, 6.04666336949399138274736569349, 6.98538681486012228863528017766, 7.71829140208469189720640830867, 8.740784641715649707885632297645, 9.122517284665421718123851433918

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