↑

Properties

Label 2-2020-2020.1347-c0-0-0
Degree $2$
Conductor $2020$
Sign $0.948 + 0.315i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.187 − 0.982i)2-s + (−0.929 − 0.368i)4-s + (−0.998 − 0.0627i)5-s + (−0.535 + 0.844i)8-s + (0.637 + 0.770i)9-s + (−0.248 + 0.968i)10-s + (0.535 + 0.415i)13-s + (0.728 + 0.684i)16-s + (−0.142 + 0.896i)17-s + (0.876 − 0.481i)18-s + (0.904 + 0.425i)20-s + (0.992 + 0.125i)25-s + (0.508 − 0.448i)26-s + (−1.21 + 1.56i)29-s + (0.809 − 0.587i)32-s + ⋯
L(s)  = 1  + (0.187 − 0.982i)2-s + (−0.929 − 0.368i)4-s + (−0.998 − 0.0627i)5-s + (−0.535 + 0.844i)8-s + (0.637 + 0.770i)9-s + (−0.248 + 0.968i)10-s + (0.535 + 0.415i)13-s + (0.728 + 0.684i)16-s + (−0.142 + 0.896i)17-s + (0.876 − 0.481i)18-s + (0.904 + 0.425i)20-s + (0.992 + 0.125i)25-s + (0.508 − 0.448i)26-s + (−1.21 + 1.56i)29-s + (0.809 − 0.587i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $0.948 + 0.315i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ 0.948 + 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9364437819\)
\(L(\frac12)\) \(\approx\) \(0.9364437819\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.187 + 0.982i)T \)
5 \( 1 + (0.998 + 0.0627i)T \)
101 \( 1 + (0.368 - 0.929i)T \)
good3 \( 1 + (-0.637 - 0.770i)T^{2} \)
7 \( 1 + (-0.968 - 0.248i)T^{2} \)
11 \( 1 + (-0.982 - 0.187i)T^{2} \)
13 \( 1 + (-0.535 - 0.415i)T + (0.248 + 0.968i)T^{2} \)
17 \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.0627 - 0.998i)T^{2} \)
23 \( 1 + (0.844 - 0.535i)T^{2} \)
29 \( 1 + (1.21 - 1.56i)T + (-0.248 - 0.968i)T^{2} \)
31 \( 1 + (-0.968 - 0.248i)T^{2} \)
37 \( 1 + (-0.269 + 0.747i)T + (-0.770 - 0.637i)T^{2} \)
41 \( 1 + (-1.97 + 0.312i)T + (0.951 - 0.309i)T^{2} \)
43 \( 1 + (-0.125 + 0.992i)T^{2} \)
47 \( 1 + (-0.125 - 0.992i)T^{2} \)
53 \( 1 + (-0.233 + 0.0922i)T + (0.728 - 0.684i)T^{2} \)
59 \( 1 + (0.998 - 0.0627i)T^{2} \)
61 \( 1 + (-0.400 + 0.173i)T + (0.684 - 0.728i)T^{2} \)
67 \( 1 + (0.637 - 0.770i)T^{2} \)
71 \( 1 + (0.637 + 0.770i)T^{2} \)
73 \( 1 + (0.0604 - 0.110i)T + (-0.535 - 0.844i)T^{2} \)
79 \( 1 + (0.535 - 0.844i)T^{2} \)
83 \( 1 + (0.535 - 0.844i)T^{2} \)
89 \( 1 + (0.188 + 0.00591i)T + (0.998 + 0.0627i)T^{2} \)
97 \( 1 + (0.486 + 1.12i)T + (-0.684 + 0.728i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.251446960569531731411229215632, −8.701362794495897275674478407122, −7.83949679330628696990292127845, −7.14411546426805524155577916192, −5.91678976843915394553323507790, −5.00225907607377854375243124576, −4.10926552574701639157134120263, −3.67982627897428963989068005297, −2.38939174982146376790507350444, −1.29848820775710152289461504679, 0.75185919227008364439839887094, 2.91341411744299002228004822938, 3.94465128340710323586384870429, 4.35557524248294373183202256961, 5.50196815553690289109764248565, 6.31041737494146255260811576058, 7.13875886203038852640818742382, 7.66953121181608729738372509461, 8.393442944259296472743879942479, 9.260647033994868378254300329607

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