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Properties

Label 2-2020-2020.1179-c0-0-1
Degree $2$
Conductor $2020$
Sign $0.410 + 0.911i$
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 + (1.23 + 0.317i)3-s + (−0.425 + 0.904i)4-s + (0.535 − 0.844i)5-s + (−0.393 − 1.21i)6-s + (0.996 − 0.394i)7-s + (0.992 − 0.125i)8-s + (0.547 + 0.301i)9-s − 10-s + (−0.812 + 0.982i)12-s + (−0.866 − 0.629i)14-s + (0.929 − 0.872i)15-s + (−0.637 − 0.770i)16-s + (−0.0392 − 0.624i)18-s + (0.535 + 0.844i)20-s + (1.35 − 0.171i)21-s + ⋯
L(s)  = 1  + (−0.535 − 0.844i)2-s + (1.23 + 0.317i)3-s + (−0.425 + 0.904i)4-s + (0.535 − 0.844i)5-s + (−0.393 − 1.21i)6-s + (0.996 − 0.394i)7-s + (0.992 − 0.125i)8-s + (0.547 + 0.301i)9-s − 10-s + (−0.812 + 0.982i)12-s + (−0.866 − 0.629i)14-s + (0.929 − 0.872i)15-s + (−0.637 − 0.770i)16-s + (−0.0392 − 0.624i)18-s + (0.535 + 0.844i)20-s + (1.35 − 0.171i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $0.410 + 0.911i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ 0.410 + 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.527511732\)
\(L(\frac12)\) \(\approx\) \(1.527511732\)
\(L(1)\) 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 \( 1 + (-0.535 + 0.844i)T \)
101 \( 1 + (0.992 - 0.125i)T \)
good3 \( 1 + (-1.23 - 0.317i)T + (0.876 + 0.481i)T^{2} \)
7 \( 1 + (-0.996 + 0.394i)T + (0.728 - 0.684i)T^{2} \)
11 \( 1 + (-0.535 - 0.844i)T^{2} \)
13 \( 1 + (-0.728 - 0.684i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.187 + 0.982i)T^{2} \)
23 \( 1 + (0.0915 - 1.45i)T + (-0.992 - 0.125i)T^{2} \)
29 \( 1 + (0.996 + 0.394i)T + (0.728 + 0.684i)T^{2} \)
31 \( 1 + (-0.728 + 0.684i)T^{2} \)
37 \( 1 + (-0.876 + 0.481i)T^{2} \)
41 \( 1 + (0.574 + 1.76i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.362 - 1.90i)T + (-0.929 + 0.368i)T^{2} \)
47 \( 1 + (0.238 - 1.25i)T + (-0.929 - 0.368i)T^{2} \)
53 \( 1 + (0.637 - 0.770i)T^{2} \)
59 \( 1 + (0.187 - 0.982i)T^{2} \)
61 \( 1 + (0.746 - 1.58i)T + (-0.637 - 0.770i)T^{2} \)
67 \( 1 + (0.121 - 0.0312i)T + (0.876 - 0.481i)T^{2} \)
71 \( 1 + (-0.876 - 0.481i)T^{2} \)
73 \( 1 + (0.992 + 0.125i)T^{2} \)
79 \( 1 + (0.992 - 0.125i)T^{2} \)
83 \( 1 + (0.121 + 1.93i)T + (-0.992 + 0.125i)T^{2} \)
89 \( 1 + (-1.03 + 1.24i)T + (-0.187 - 0.982i)T^{2} \)
97 \( 1 + (0.637 + 0.770i)T^{2} \)
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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

−9.149884829827436477682885740861, −8.712156384597866710064128018777, −7.77503459551765461962500288818, −7.57170825112014059104312332060, −5.84702970695303110706284649276, −4.77602947400703371314531428872, −4.10464576939893301879362440335, −3.20855956188142235850300806323, −2.10693939956084880226152719486, −1.38334294222204693934938332744, 1.74093742734971947389226975098, 2.35671601692971776748295908829, 3.56932702012664052562909656083, 4.83586180535784346711335055982, 5.65319371671492006569816256657, 6.61558608440479367859177239309, 7.25079353940069725493917315585, 8.075788790952605602717326088187, 8.496315815101732615419147338428, 9.228984904566711331915294549588

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