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Properties

Label 2-2020-2020.1247-c0-0-0
Degree $2$
Conductor $2020$
Sign $-0.804 + 0.594i$
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  + (0.248 − 0.968i)2-s + (−0.876 − 0.481i)4-s + (0.998 − 0.0627i)5-s + (−0.684 + 0.728i)8-s + (−0.992 − 0.125i)9-s + (0.187 − 0.982i)10-s + (0.141 − 1.49i)13-s + (0.535 + 0.844i)16-s + (−1.76 − 0.896i)17-s + (−0.368 + 0.929i)18-s + (−0.904 − 0.425i)20-s + (0.992 − 0.125i)25-s + (−1.41 − 0.508i)26-s + (0.167 − 1.77i)29-s + (0.951 − 0.309i)32-s + ⋯
L(s)  = 1  + (0.248 − 0.968i)2-s + (−0.876 − 0.481i)4-s + (0.998 − 0.0627i)5-s + (−0.684 + 0.728i)8-s + (−0.992 − 0.125i)9-s + (0.187 − 0.982i)10-s + (0.141 − 1.49i)13-s + (0.535 + 0.844i)16-s + (−1.76 − 0.896i)17-s + (−0.368 + 0.929i)18-s + (−0.904 − 0.425i)20-s + (0.992 − 0.125i)25-s + (−1.41 − 0.508i)26-s + (0.167 − 1.77i)29-s + (0.951 − 0.309i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $-0.804 + 0.594i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ -0.804 + 0.594i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.137955277\)
\(L(\frac12)\) \(\approx\) \(1.137955277\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.248 + 0.968i)T \)
5 \( 1 + (-0.998 + 0.0627i)T \)
101 \( 1 + (0.481 - 0.876i)T \)
good3 \( 1 + (0.992 + 0.125i)T^{2} \)
7 \( 1 + (-0.187 - 0.982i)T^{2} \)
11 \( 1 + (0.248 - 0.968i)T^{2} \)
13 \( 1 + (-0.141 + 1.49i)T + (-0.982 - 0.187i)T^{2} \)
17 \( 1 + (1.76 + 0.896i)T + (0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.425 - 0.904i)T^{2} \)
23 \( 1 + (-0.684 - 0.728i)T^{2} \)
29 \( 1 + (-0.167 + 1.77i)T + (-0.982 - 0.187i)T^{2} \)
31 \( 1 + (0.187 + 0.982i)T^{2} \)
37 \( 1 + (-1.27 + 1.44i)T + (-0.125 - 0.992i)T^{2} \)
41 \( 1 + (-0.388 + 0.198i)T + (0.587 - 0.809i)T^{2} \)
43 \( 1 + (0.770 + 0.637i)T^{2} \)
47 \( 1 + (0.770 - 0.637i)T^{2} \)
53 \( 1 + (-0.742 - 1.35i)T + (-0.535 + 0.844i)T^{2} \)
59 \( 1 + (-0.904 - 0.425i)T^{2} \)
61 \( 1 + (-0.0175 - 0.0603i)T + (-0.844 + 0.535i)T^{2} \)
67 \( 1 + (-0.992 + 0.125i)T^{2} \)
71 \( 1 + (0.992 + 0.125i)T^{2} \)
73 \( 1 + (0.791 - 0.313i)T + (0.728 - 0.684i)T^{2} \)
79 \( 1 + (0.728 + 0.684i)T^{2} \)
83 \( 1 + (-0.728 - 0.684i)T^{2} \)
89 \( 1 + (-0.344 - 1.54i)T + (-0.904 + 0.425i)T^{2} \)
97 \( 1 + (0.0525 + 0.180i)T + (-0.844 + 0.535i)T^{2} \)
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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

−9.190272606999458538913420146639, −8.595643096581282662562103608475, −7.63329279405840654390439296664, −6.21016904266843646273574222299, −5.80165980283473222792266541934, −4.96545844819144251067466465375, −4.02615731892306700290029752371, −2.58407452097981754689530259322, −2.53382542262471062509788278479, −0.74083276988135643204143227235, 1.84360208815361146598007276005, 2.98293007525671695168234325057, 4.20454843047275794945170368364, 4.94905748976634651926236371695, 5.83106634206927682767447038467, 6.57119450746124587936252107098, 6.89349765129466757273336921967, 8.267284366663335141403323644681, 8.820052474403007944790323896244, 9.260169695604011702153352164409

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