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Properties

Label 2-2020-2020.1299-c0-0-1
Degree $2$
Conductor $2020$
Sign $0.984 - 0.176i$
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  + (−0.309 + 0.951i)2-s + (−0.190 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + 0.618·6-s + (−0.190 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.5 − 0.363i)9-s − 0.999·10-s + (−0.190 + 0.587i)12-s + 0.618·14-s + (0.5 − 0.363i)15-s + (0.309 + 0.951i)16-s + (0.190 + 0.587i)18-s + (0.309 − 0.951i)20-s + (−0.309 + 0.224i)21-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.190 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + 0.618·6-s + (−0.190 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.5 − 0.363i)9-s − 0.999·10-s + (−0.190 + 0.587i)12-s + 0.618·14-s + (0.5 − 0.363i)15-s + (0.309 + 0.951i)16-s + (0.190 + 0.587i)18-s + (0.309 − 0.951i)20-s + (−0.309 + 0.224i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $0.984 - 0.176i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ 0.984 - 0.176i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9195611500\)
\(L(\frac12)\) \(\approx\) \(0.9195611500\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
101 \( 1 + (0.809 - 0.587i)T \)
good3 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
47 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−9.309268223543430988738663270112, −8.422313913774136243260869860963, −7.43847811017561345810776560270, −7.06735863285583340622975513296, −6.38571053213904893411660141156, −5.81376377429101635365119230354, −4.56207905221726940338786319165, −3.74583497681632454405118621740, −2.35012398547061609002614945865, −0.902477680101409241382785086161, 1.28227800709021685835594161377, 2.30009490288539903514911082172, 3.51779419432741888798749399316, 4.36815820267629233361795972937, 5.15318917295852498345306227092, 5.71560281506232263721308000836, 7.20367567129793366174443371035, 8.044081358232440494399383378984, 8.885217516841150052451543081259, 9.453472669853455939549655833918

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