Properties

Label 2-2020-2020.1259-c0-0-0
Degree $2$
Conductor $2020$
Sign $-0.781 - 0.624i$
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 + (−1.06 − 0.0672i)3-s + (−0.876 − 0.481i)4-s + (−0.968 + 0.248i)5-s + (0.331 − 1.01i)6-s + (1.49 + 1.23i)7-s + (0.684 − 0.728i)8-s + (0.147 + 0.0186i)9-s i·10-s + (0.904 + 0.574i)12-s + (−1.56 + 1.13i)14-s + (1.05 − 0.200i)15-s + (0.535 + 0.844i)16-s + (−0.0546 + 0.138i)18-s + (0.968 + 0.248i)20-s + (−1.51 − 1.42i)21-s + ⋯
L(s)  = 1  + (−0.248 + 0.968i)2-s + (−1.06 − 0.0672i)3-s + (−0.876 − 0.481i)4-s + (−0.968 + 0.248i)5-s + (0.331 − 1.01i)6-s + (1.49 + 1.23i)7-s + (0.684 − 0.728i)8-s + (0.147 + 0.0186i)9-s i·10-s + (0.904 + 0.574i)12-s + (−1.56 + 1.13i)14-s + (1.05 − 0.200i)15-s + (0.535 + 0.844i)16-s + (−0.0546 + 0.138i)18-s + (0.968 + 0.248i)20-s + (−1.51 − 1.42i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $-0.781 - 0.624i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ -0.781 - 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5534800173\)
\(L(\frac12)\) \(\approx\) \(0.5534800173\)
\(L(1)\) 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 \( 1 + (0.968 - 0.248i)T \)
101 \( 1 + (0.728 + 0.684i)T \)
good3 \( 1 + (1.06 + 0.0672i)T + (0.992 + 0.125i)T^{2} \)
7 \( 1 + (-1.49 - 1.23i)T + (0.187 + 0.982i)T^{2} \)
11 \( 1 + (0.968 + 0.248i)T^{2} \)
13 \( 1 + (0.187 - 0.982i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.425 + 0.904i)T^{2} \)
23 \( 1 + (-1.82 + 0.723i)T + (0.728 - 0.684i)T^{2} \)
29 \( 1 + (0.383 - 0.317i)T + (0.187 - 0.982i)T^{2} \)
31 \( 1 + (0.187 + 0.982i)T^{2} \)
37 \( 1 + (0.992 - 0.125i)T^{2} \)
41 \( 1 + (1.46 + 0.476i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.849 - 1.80i)T + (-0.637 + 0.770i)T^{2} \)
47 \( 1 + (0.718 - 1.52i)T + (-0.637 - 0.770i)T^{2} \)
53 \( 1 + (0.535 - 0.844i)T^{2} \)
59 \( 1 + (-0.425 + 0.904i)T^{2} \)
61 \( 1 + (0.120 - 0.219i)T + (-0.535 - 0.844i)T^{2} \)
67 \( 1 + (1.85 - 0.116i)T + (0.992 - 0.125i)T^{2} \)
71 \( 1 + (0.992 + 0.125i)T^{2} \)
73 \( 1 + (0.728 - 0.684i)T^{2} \)
79 \( 1 + (-0.728 - 0.684i)T^{2} \)
83 \( 1 + (-0.0462 + 0.116i)T + (-0.728 - 0.684i)T^{2} \)
89 \( 1 + (-0.992 - 0.629i)T + (0.425 + 0.904i)T^{2} \)
97 \( 1 + (-0.535 - 0.844i)T^{2} \)
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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

−9.279206352278054967557566318125, −8.655369822147970152182906330593, −8.060941855766392122110305138925, −7.30071501516127664811464178406, −6.46022835139943400920888704632, −5.69597500641542397699623313956, −4.88474170223742599376466029008, −4.58483018171897157098674420564, −2.96042005876533082401620096982, −1.24408373051907785165170688318, 0.61740063057727790017267552833, 1.61279337559770074630136224081, 3.29832011651553375006780251349, 4.15421136224161479385215905364, 4.91676580666230521166396499974, 5.30695690726975953745555156034, 6.99282706143752495487617786525, 7.52462417413920968029349352868, 8.357170236187871633868426632989, 8.952180637430091095629893308578

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