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Properties

Label 2-2020-2020.2019-c0-0-16
Degree $2$
Conductor $2020$
Sign $i$
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  + (0.707 − 0.707i)2-s + 1.41i·3-s − 1.00i·4-s − 5-s + (1.00 + 1.00i)6-s − 1.41i·7-s + (−0.707 − 0.707i)8-s − 1.00·9-s + (−0.707 + 0.707i)10-s + 1.41·12-s + (−1.00 − 1.00i)14-s − 1.41i·15-s − 1.00·16-s + (−0.707 + 0.707i)18-s − 2i·19-s + 1.00i·20-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + 1.41i·3-s − 1.00i·4-s − 5-s + (1.00 + 1.00i)6-s − 1.41i·7-s + (−0.707 − 0.707i)8-s − 1.00·9-s + (−0.707 + 0.707i)10-s + 1.41·12-s + (−1.00 − 1.00i)14-s − 1.41i·15-s − 1.00·16-s + (−0.707 + 0.707i)18-s − 2i·19-s + 1.00i·20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (2019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.195914410\)
\(L(\frac12)\) \(\approx\) \(1.195914410\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + T \)
101 \( 1 - T \)
good3 \( 1 - 1.41iT - T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 2iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 2iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.41T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.417073663750510845442906153102, −8.662459065926336787991645797164, −7.43122099073419260058304912801, −6.81900239307235069375458153382, −5.53506008074771268021375800029, −4.54210615549178868434633730857, −4.28821018129192184129566041792, −3.60738468724815490147735045445, −2.68833249922696333081740059644, −0.69136542285552022293234813200, 1.74662287686896070331652727446, 2.85094934261639941975794383245, 3.70453628238818356272461205292, 4.91234262449268314522222221152, 5.77657847698702990275298694623, 6.37155197638990928303065105770, 7.20719595088270669106977831270, 7.81232696845003102319129687186, 8.459214536838837449839345094695, 8.914182051508862261689389142979

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