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Properties

Label 2-2020-2020.1059-c0-0-0
Degree $2$
Conductor $2020$
Sign $-0.841 - 0.539i$
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  + (0.844 + 0.535i)2-s + (0.317 + 1.23i)3-s + (0.425 + 0.904i)4-s + (−0.535 − 0.844i)5-s + (−0.393 + 1.21i)6-s + (−0.394 + 0.996i)7-s + (−0.125 + 0.992i)8-s + (−0.547 + 0.301i)9-s − i·10-s + (−0.982 + 0.812i)12-s + (−0.866 + 0.629i)14-s + (0.872 − 0.929i)15-s + (−0.637 + 0.770i)16-s + (−0.624 − 0.0392i)18-s + (0.535 − 0.844i)20-s + (−1.35 − 0.171i)21-s + ⋯
L(s)  = 1  + (0.844 + 0.535i)2-s + (0.317 + 1.23i)3-s + (0.425 + 0.904i)4-s + (−0.535 − 0.844i)5-s + (−0.393 + 1.21i)6-s + (−0.394 + 0.996i)7-s + (−0.125 + 0.992i)8-s + (−0.547 + 0.301i)9-s − i·10-s + (−0.982 + 0.812i)12-s + (−0.866 + 0.629i)14-s + (0.872 − 0.929i)15-s + (−0.637 + 0.770i)16-s + (−0.624 − 0.0392i)18-s + (0.535 − 0.844i)20-s + (−1.35 − 0.171i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $-0.841 - 0.539i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ -0.841 - 0.539i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.761145612\)
\(L(\frac12)\) \(\approx\) \(1.761145612\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.844 - 0.535i)T \)
5 \( 1 + (0.535 + 0.844i)T \)
101 \( 1 + (-0.992 - 0.125i)T \)
good3 \( 1 + (-0.317 - 1.23i)T + (-0.876 + 0.481i)T^{2} \)
7 \( 1 + (0.394 - 0.996i)T + (-0.728 - 0.684i)T^{2} \)
11 \( 1 + (0.535 - 0.844i)T^{2} \)
13 \( 1 + (-0.728 + 0.684i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.187 - 0.982i)T^{2} \)
23 \( 1 + (-0.0859 - 1.36i)T + (-0.992 + 0.125i)T^{2} \)
29 \( 1 + (0.621 + 1.57i)T + (-0.728 + 0.684i)T^{2} \)
31 \( 1 + (-0.728 - 0.684i)T^{2} \)
37 \( 1 + (-0.876 - 0.481i)T^{2} \)
41 \( 1 + (-0.700 - 0.227i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.0931 + 0.488i)T + (-0.929 - 0.368i)T^{2} \)
47 \( 1 + (0.288 + 1.51i)T + (-0.929 + 0.368i)T^{2} \)
53 \( 1 + (-0.637 - 0.770i)T^{2} \)
59 \( 1 + (-0.187 - 0.982i)T^{2} \)
61 \( 1 + (-0.871 + 0.410i)T + (0.637 - 0.770i)T^{2} \)
67 \( 1 + (-0.0312 + 0.121i)T + (-0.876 - 0.481i)T^{2} \)
71 \( 1 + (-0.876 + 0.481i)T^{2} \)
73 \( 1 + (-0.992 + 0.125i)T^{2} \)
79 \( 1 + (0.992 + 0.125i)T^{2} \)
83 \( 1 + (-1.93 - 0.121i)T + (0.992 + 0.125i)T^{2} \)
89 \( 1 + (-0.905 + 0.749i)T + (0.187 - 0.982i)T^{2} \)
97 \( 1 + (0.637 - 0.770i)T^{2} \)
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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

−9.322767617460661944199682349879, −8.992322773672379267537023693196, −8.125093950355677143124788491541, −7.41228199056203215684423886328, −6.19627734782983736977102120519, −5.41699255265516369176303945435, −4.86457393107338839433085118978, −3.90672748064650684297188351876, −3.46129712068513839125469653566, −2.24439781666772773232930415504, 0.948549536737288858292795218845, 2.21334424446170058609259699275, 3.05644378228248294627407476123, 3.86745966528636939854538258041, 4.76516017506085281620888681564, 6.13279139139849823879923324598, 6.70308758159782431028055592126, 7.25088581887116291097915338223, 7.86841792810336786756483583560, 9.016755794646621853943738589705

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