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Properties

Label 2-2020-2020.1267-c0-0-0
Degree $2$
Conductor $2020$
Sign $-0.509 - 0.860i$
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  + (−0.368 + 0.929i)2-s + (−0.728 − 0.684i)4-s + (0.770 + 0.637i)5-s + (0.904 − 0.425i)8-s + (−0.187 + 0.982i)9-s + (−0.876 + 0.481i)10-s + (0.162 + 0.0958i)13-s + (0.0627 + 0.998i)16-s + (0.142 + 0.278i)17-s + (−0.844 − 0.535i)18-s + (−0.125 − 0.992i)20-s + (0.187 + 0.982i)25-s + (−0.148 + 0.115i)26-s + (−0.781 − 0.462i)29-s + (−0.951 − 0.309i)32-s + ⋯
L(s)  = 1  + (−0.368 + 0.929i)2-s + (−0.728 − 0.684i)4-s + (0.770 + 0.637i)5-s + (0.904 − 0.425i)8-s + (−0.187 + 0.982i)9-s + (−0.876 + 0.481i)10-s + (0.162 + 0.0958i)13-s + (0.0627 + 0.998i)16-s + (0.142 + 0.278i)17-s + (−0.844 − 0.535i)18-s + (−0.125 − 0.992i)20-s + (0.187 + 0.982i)25-s + (−0.148 + 0.115i)26-s + (−0.781 − 0.462i)29-s + (−0.951 − 0.309i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $-0.509 - 0.860i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ -0.509 - 0.860i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9868792409\)
\(L(\frac12)\) \(\approx\) \(0.9868792409\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.368 - 0.929i)T \)
5 \( 1 + (-0.770 - 0.637i)T \)
101 \( 1 + (0.684 - 0.728i)T \)
good3 \( 1 + (0.187 - 0.982i)T^{2} \)
7 \( 1 + (0.876 + 0.481i)T^{2} \)
11 \( 1 + (-0.368 + 0.929i)T^{2} \)
13 \( 1 + (-0.162 - 0.0958i)T + (0.481 + 0.876i)T^{2} \)
17 \( 1 + (-0.142 - 0.278i)T + (-0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.992 - 0.125i)T^{2} \)
23 \( 1 + (0.904 + 0.425i)T^{2} \)
29 \( 1 + (0.781 + 0.462i)T + (0.481 + 0.876i)T^{2} \)
31 \( 1 + (-0.876 - 0.481i)T^{2} \)
37 \( 1 + (-1.99 + 0.188i)T + (0.982 - 0.187i)T^{2} \)
41 \( 1 + (0.600 - 1.17i)T + (-0.587 - 0.809i)T^{2} \)
43 \( 1 + (0.248 - 0.968i)T^{2} \)
47 \( 1 + (0.248 + 0.968i)T^{2} \)
53 \( 1 + (-0.340 - 0.362i)T + (-0.0627 + 0.998i)T^{2} \)
59 \( 1 + (-0.125 - 0.992i)T^{2} \)
61 \( 1 + (-0.0212 - 0.677i)T + (-0.998 + 0.0627i)T^{2} \)
67 \( 1 + (-0.187 - 0.982i)T^{2} \)
71 \( 1 + (0.187 - 0.982i)T^{2} \)
73 \( 1 + (-1.06 - 1.67i)T + (-0.425 + 0.904i)T^{2} \)
79 \( 1 + (-0.425 - 0.904i)T^{2} \)
83 \( 1 + (0.425 + 0.904i)T^{2} \)
89 \( 1 + (0.743 + 0.843i)T + (-0.125 + 0.992i)T^{2} \)
97 \( 1 + (0.0540 + 1.72i)T + (-0.998 + 0.0627i)T^{2} \)
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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

−9.683040033750969042848922044438, −8.694417755935789936777381645902, −7.950197588919063915037202295226, −7.30270673191468710790189131705, −6.41447589406469662856933481549, −5.80344849299408901346152441999, −5.06526627437563983980223984457, −4.06643186801550511023127787987, −2.69026350304259171097411195338, −1.58401881054891117703851652209, 0.866215637845801388923974768561, 1.98508450077732944770504870644, 3.06258168146082962370678481327, 3.97371043184108829869302436923, 4.93344062295546657923889763852, 5.77565609380990928746992130869, 6.70664687115704103281993649479, 7.81286842126577252021276005880, 8.566552563568465298765693827743, 9.390047835151181293892969668526

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