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Properties

Label 2-2020-2020.1319-c0-0-0
Degree $2$
Conductor $2020$
Sign $0.0732 - 0.997i$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)10-s + (−0.690 + 0.951i)13-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)18-s + (−0.309 + 0.951i)20-s + (0.309 + 0.951i)25-s + (−1.11 + 0.363i)26-s − 32-s + 36-s + (1.11 − 1.53i)37-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)10-s + (−0.690 + 0.951i)13-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)18-s + (−0.309 + 0.951i)20-s + (0.309 + 0.951i)25-s + (−1.11 + 0.363i)26-s − 32-s + 36-s + (1.11 − 1.53i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0732 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0732 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $0.0732 - 0.997i$
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: \((2,\ 2020,\ (\ :0),\ 0.0732 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.045050425\)
\(L(\frac12)\) \(\approx\) \(2.045050425\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
101 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
11 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + 1.17iT - T^{2} \)
43 \( 1 + (-0.809 + 0.587i)T^{2} \)
47 \( 1 + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
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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

−9.291125131039435901478139086892, −8.920229441265060248951810919257, −7.50304754676637373158975047917, −7.11678283152177614608362831966, −6.25186395204221706691126325747, −5.76678111784615229315192660518, −4.66942892485123483410492574229, −3.87750536983018381855519495775, −2.88179984850191193189637032221, −1.93093858056136446531024939623, 1.27375540719858488911958199381, 2.30999938191714398519975768047, 3.11546650886380019864369890293, 4.55605335948003284878690166038, 4.91945806086055070800053091450, 5.76000483721449293874416180945, 6.49540779955646628793901871409, 7.59910708741433215052784568638, 8.367327248820095453391446377388, 9.568839707935530970100258519368

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