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Properties

Label 2-2020-2020.1963-c0-0-0
Degree $2$
Conductor $2020$
Sign $0.786 - 0.618i$
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  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.951 + 0.309i)5-s + (−0.587 + 0.809i)8-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)10-s + (−0.809 − 0.412i)13-s + (0.309 − 0.951i)16-s + (1 + i)17-s + (0.951 + 0.309i)18-s + (−0.587 + 0.809i)20-s + (0.809 − 0.587i)25-s + (0.896 + 0.142i)26-s + (1.26 + 0.642i)29-s + i·32-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.951 + 0.309i)5-s + (−0.587 + 0.809i)8-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)10-s + (−0.809 − 0.412i)13-s + (0.309 − 0.951i)16-s + (1 + i)17-s + (0.951 + 0.309i)18-s + (−0.587 + 0.809i)20-s + (0.809 − 0.587i)25-s + (0.896 + 0.142i)26-s + (1.26 + 0.642i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $0.786 - 0.618i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1963, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ 0.786 - 0.618i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5524268480\)
\(L(\frac12)\) \(\approx\) \(0.5524268480\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 + (0.951 - 0.309i)T \)
101 \( 1 + (-0.587 - 0.809i)T \)
good3 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T^{2} \)
11 \( 1 + (-0.951 + 0.309i)T^{2} \)
13 \( 1 + (0.809 + 0.412i)T + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T^{2} \)
29 \( 1 + (-1.26 - 0.642i)T + (0.587 + 0.809i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-1.76 - 0.896i)T + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
43 \( 1 + (0.951 + 0.309i)T^{2} \)
47 \( 1 + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.587 + 0.809i)T^{2} \)
61 \( 1 + (-0.309 + 1.95i)T + (-0.951 - 0.309i)T^{2} \)
67 \( 1 + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.142 + 0.278i)T + (-0.587 - 0.809i)T^{2} \)
97 \( 1 + (-0.278 + 1.76i)T + (-0.951 - 0.309i)T^{2} \)
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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

−9.367027326697102951582275972662, −8.428590428719309635664629311675, −7.998970680341027047028511260947, −7.28742056827287197167264938978, −6.39398192835596235125903095131, −5.73604124946635404973308200662, −4.61763912332846693094696778676, −3.34033845395688292660633607539, −2.63501884533264461977747711657, −0.934929223742344176396251079819, 0.74070881818637227833923920115, 2.37953637921410358420993284096, 3.10781632184058405383054814121, 4.26646525039635060832854313998, 5.18656111448030006978263432890, 6.28788833810262758070083118724, 7.34810872565044678799282394261, 7.77616201865230149770242389238, 8.429233833959567434906836919495, 9.252642189535031805998064019923

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