↑

Properties

Label 2-2020-2020.1083-c0-0-0
Degree $2$
Conductor $2020$
Sign $-0.281 - 0.959i$
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  + (0.425 + 0.904i)2-s + (−0.637 + 0.770i)4-s + (0.368 − 0.929i)5-s + (−0.968 − 0.248i)8-s + (−0.535 + 0.844i)9-s + (0.998 − 0.0627i)10-s + (−0.0603 + 1.91i)13-s + (−0.187 − 0.982i)16-s + (1.76 + 0.278i)17-s + (−0.992 − 0.125i)18-s + (0.481 + 0.876i)20-s + (−0.728 − 0.684i)25-s + (−1.76 + 0.762i)26-s + (0.312 + 0.00982i)29-s + (0.809 − 0.587i)32-s + ⋯
L(s)  = 1  + (0.425 + 0.904i)2-s + (−0.637 + 0.770i)4-s + (0.368 − 0.929i)5-s + (−0.968 − 0.248i)8-s + (−0.535 + 0.844i)9-s + (0.998 − 0.0627i)10-s + (−0.0603 + 1.91i)13-s + (−0.187 − 0.982i)16-s + (1.76 + 0.278i)17-s + (−0.992 − 0.125i)18-s + (0.481 + 0.876i)20-s + (−0.728 − 0.684i)25-s + (−1.76 + 0.762i)26-s + (0.312 + 0.00982i)29-s + (0.809 − 0.587i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $-0.281 - 0.959i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1083, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ -0.281 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.334068813\)
\(L(\frac12)\) \(\approx\) \(1.334068813\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.425 - 0.904i)T \)
5 \( 1 + (-0.368 + 0.929i)T \)
101 \( 1 + (0.770 + 0.637i)T \)
good3 \( 1 + (0.535 - 0.844i)T^{2} \)
7 \( 1 + (-0.0627 - 0.998i)T^{2} \)
11 \( 1 + (-0.904 + 0.425i)T^{2} \)
13 \( 1 + (0.0603 - 1.91i)T + (-0.998 - 0.0627i)T^{2} \)
17 \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \)
19 \( 1 + (-0.929 - 0.368i)T^{2} \)
23 \( 1 + (0.248 + 0.968i)T^{2} \)
29 \( 1 + (-0.312 - 0.00982i)T + (0.998 + 0.0627i)T^{2} \)
31 \( 1 + (-0.0627 - 0.998i)T^{2} \)
37 \( 1 + (0.180 - 0.0525i)T + (0.844 - 0.535i)T^{2} \)
41 \( 1 + (-0.175 - 1.11i)T + (-0.951 + 0.309i)T^{2} \)
43 \( 1 + (-0.684 + 0.728i)T^{2} \)
47 \( 1 + (-0.684 - 0.728i)T^{2} \)
53 \( 1 + (-0.872 - 1.05i)T + (-0.187 + 0.982i)T^{2} \)
59 \( 1 + (-0.368 - 0.929i)T^{2} \)
61 \( 1 + (1.01 - 0.0958i)T + (0.982 - 0.187i)T^{2} \)
67 \( 1 + (-0.535 - 0.844i)T^{2} \)
71 \( 1 + (-0.535 + 0.844i)T^{2} \)
73 \( 1 + (0.233 + 1.84i)T + (-0.968 + 0.248i)T^{2} \)
79 \( 1 + (0.968 + 0.248i)T^{2} \)
83 \( 1 + (0.968 + 0.248i)T^{2} \)
89 \( 1 + (-0.245 - 0.360i)T + (-0.368 + 0.929i)T^{2} \)
97 \( 1 + (0.188 + 1.99i)T + (-0.982 + 0.187i)T^{2} \)
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

−9.319082418410697267343621334265, −8.679268507645763348497631439902, −7.966341719725976454246921456430, −7.28673948141286929867124277556, −6.20265048747807693690205768701, −5.65584128607385975904114718269, −4.77024050364883950503290970506, −4.22839758963564449113531364042, −2.96584407841999789638288616394, −1.60425831095660438544106419871, 0.903281507490004339151177171794, 2.46722651740564004391654121978, 3.23752777822804776117341548297, 3.69224598904308603946796038284, 5.37081243425096292301269388318, 5.58571314916233534481892195819, 6.53004179464434643252834390200, 7.58957812709435689684821309465, 8.440329423919522765672974892355, 9.411453448744890711531027403091

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