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Properties

Label 2-2020-2020.1007-c0-0-0
Degree $2$
Conductor $2020$
Sign $0.809 - 0.586i$
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  + (0.187 + 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 + (−1.15 − 1.48i)13-s + (0.728 − 0.684i)16-s + (1.76 − 0.278i)17-s + (0.876 + 0.481i)18-s + (−0.904 + 0.425i)20-s + (0.992 − 0.125i)25-s + (1.24 − 1.41i)26-s + (−0.247 + 0.191i)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 + (−1.15 − 1.48i)13-s + (0.728 − 0.684i)16-s + (1.76 − 0.278i)17-s + (0.876 + 0.481i)18-s + (−0.904 + 0.425i)20-s + (0.992 − 0.125i)25-s + (1.24 − 1.41i)26-s + (−0.247 + 0.191i)29-s + (0.809 + 0.587i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $0.809 - 0.586i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ 0.809 - 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.398180824\)
\(L(\frac12)\) \(\approx\) \(1.398180824\)
\(L(1)\) 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 \( 1 + (1.15 + 1.48i)T + (-0.248 + 0.968i)T^{2} \)
17 \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.0627 + 0.998i)T^{2} \)
23 \( 1 + (-0.844 - 0.535i)T^{2} \)
29 \( 1 + (0.247 - 0.191i)T + (0.248 - 0.968i)T^{2} \)
31 \( 1 + (-0.968 + 0.248i)T^{2} \)
37 \( 1 + (1.72 - 0.621i)T + (0.770 - 0.637i)T^{2} \)
41 \( 1 + (-0.00982 + 0.0620i)T + (-0.951 - 0.309i)T^{2} \)
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 \( 1 + (0.775 - 1.79i)T + (-0.684 - 0.728i)T^{2} \)
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 \( 1 + (-0.0625 - 1.99i)T + (-0.998 + 0.0627i)T^{2} \)
97 \( 1 + (1.45 + 0.627i)T + (0.684 + 0.728i)T^{2} \)
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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

−9.393023431018434805316908386969, −8.594535119153925053050518057815, −7.61034840366599879217449830810, −7.14293173617294648320745259252, −6.18237737773930045006955005797, −5.42173651328026484188505447873, −4.99554999357632718081071171072, −3.66898283492072152012925795408, −2.83920778750217379633685879321, −1.09172909770263327435573625904, 1.58339737470997182385196168935, 2.13911169010088334000923558031, 3.27337898992391901564904190063, 4.38767898539947777976318850840, 5.10520950932141614384660814751, 5.79762843844142955814871249281, 6.92504448534380020893104914564, 7.72757658577837050242896780271, 8.820763291552302572968175965742, 9.546796554700699293502422743042

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