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Properties

Label 2-2020-2020.1319-c0-0-1
Degree $2$
Conductor $2020$
Sign $-0.526 - 0.849i$
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  + (−0.587 + 0.809i)2-s + (0.951 + 1.30i)3-s + (−0.309 − 0.951i)4-s + (0.809 − 0.587i)5-s − 1.61·6-s + (0.951 + 1.30i)7-s + (0.951 + 0.309i)8-s + (−0.500 + 1.53i)9-s + i·10-s + (0.951 − 1.30i)12-s − 1.61·14-s + (1.53 + 0.5i)15-s + (−0.809 + 0.587i)16-s + (−0.951 − 1.30i)18-s + (−0.809 − 0.587i)20-s + (−0.809 + 2.48i)21-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (0.951 + 1.30i)3-s + (−0.309 − 0.951i)4-s + (0.809 − 0.587i)5-s − 1.61·6-s + (0.951 + 1.30i)7-s + (0.951 + 0.309i)8-s + (−0.500 + 1.53i)9-s + i·10-s + (0.951 − 1.30i)12-s − 1.61·14-s + (1.53 + 0.5i)15-s + (−0.809 + 0.587i)16-s + (−0.951 − 1.30i)18-s + (−0.809 − 0.587i)20-s + (−0.809 + 2.48i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $-0.526 - 0.849i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ -0.526 - 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.443625513\)
\(L(\frac12)\) \(\approx\) \(1.443625513\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
101 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
11 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + 1.17iT - T^{2} \)
43 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
47 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−9.340274855371492097907508680318, −8.681797458920679299651327057924, −8.511060449912589108477842345879, −7.67942912972709853900288911144, −6.17557063572388148621020139998, −5.64008516241462050587187656167, −4.76334097619153255133916163805, −4.28714118245211489016312736724, −2.59262476562631957980998528007, −1.86236259247886793780331783414, 1.38167484183406932337113724870, 1.76519067072621406043153176508, 2.88353981453734775859917930283, 3.66687723023963438185515916178, 4.82540653491857182854967952691, 6.31773361142504484903603472893, 7.06772031721490751746944964146, 7.75964990586765557414739459940, 8.085652264110330441039279027926, 9.055783935136575868631922688530

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