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Properties

Label 2-2020-2020.1259-c0-0-3
Degree $2$
Conductor $2020$
Sign $-0.781 - 0.624i$
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  + (0.248 − 0.968i)2-s + (1.06 + 0.0672i)3-s + (−0.876 − 0.481i)4-s + (−0.968 + 0.248i)5-s + (0.331 − 1.01i)6-s + (−1.49 − 1.23i)7-s + (−0.684 + 0.728i)8-s + (0.147 + 0.0186i)9-s + i·10-s + (−0.904 − 0.574i)12-s + (−1.56 + 1.13i)14-s + (−1.05 + 0.200i)15-s + (0.535 + 0.844i)16-s + (0.0546 − 0.138i)18-s + (0.968 + 0.248i)20-s + (−1.51 − 1.42i)21-s + ⋯
L(s)  = 1  + (0.248 − 0.968i)2-s + (1.06 + 0.0672i)3-s + (−0.876 − 0.481i)4-s + (−0.968 + 0.248i)5-s + (0.331 − 1.01i)6-s + (−1.49 − 1.23i)7-s + (−0.684 + 0.728i)8-s + (0.147 + 0.0186i)9-s + i·10-s + (−0.904 − 0.574i)12-s + (−1.56 + 1.13i)14-s + (−1.05 + 0.200i)15-s + (0.535 + 0.844i)16-s + (0.0546 − 0.138i)18-s + (0.968 + 0.248i)20-s + (−1.51 − 1.42i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $-0.781 - 0.624i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ -0.781 - 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4419233353\)
\(L(\frac12)\) \(\approx\) \(0.4419233353\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.248 + 0.968i)T \)
5 \( 1 + (0.968 - 0.248i)T \)
101 \( 1 + (0.728 + 0.684i)T \)
good3 \( 1 + (-1.06 - 0.0672i)T + (0.992 + 0.125i)T^{2} \)
7 \( 1 + (1.49 + 1.23i)T + (0.187 + 0.982i)T^{2} \)
11 \( 1 + (0.968 + 0.248i)T^{2} \)
13 \( 1 + (0.187 - 0.982i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.425 + 0.904i)T^{2} \)
23 \( 1 + (1.82 - 0.723i)T + (0.728 - 0.684i)T^{2} \)
29 \( 1 + (0.383 - 0.317i)T + (0.187 - 0.982i)T^{2} \)
31 \( 1 + (0.187 + 0.982i)T^{2} \)
37 \( 1 + (0.992 - 0.125i)T^{2} \)
41 \( 1 + (1.46 + 0.476i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.849 + 1.80i)T + (-0.637 + 0.770i)T^{2} \)
47 \( 1 + (-0.718 + 1.52i)T + (-0.637 - 0.770i)T^{2} \)
53 \( 1 + (0.535 - 0.844i)T^{2} \)
59 \( 1 + (-0.425 + 0.904i)T^{2} \)
61 \( 1 + (0.120 - 0.219i)T + (-0.535 - 0.844i)T^{2} \)
67 \( 1 + (-1.85 + 0.116i)T + (0.992 - 0.125i)T^{2} \)
71 \( 1 + (0.992 + 0.125i)T^{2} \)
73 \( 1 + (0.728 - 0.684i)T^{2} \)
79 \( 1 + (-0.728 - 0.684i)T^{2} \)
83 \( 1 + (0.0462 - 0.116i)T + (-0.728 - 0.684i)T^{2} \)
89 \( 1 + (-0.992 - 0.629i)T + (0.425 + 0.904i)T^{2} \)
97 \( 1 + (-0.535 - 0.844i)T^{2} \)
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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

−8.925573698323362638285283202591, −8.283596059133403824156181762509, −7.42053507205786921338313578456, −6.66432243702043992709146766905, −5.51136559708096939436541890361, −4.09858174292479975664025685409, −3.65566356133982832706719562881, −3.25346703469408689780558954310, −2.07634171582965392318725163598, −0.23892730732102491018306938851, 2.49613917221856437888596736180, 3.31376035365357943041459741778, 3.92650349019422074083302966507, 5.05075482976637207539875564946, 6.09459260758361186478519205295, 6.57090901023936935441939967496, 7.74145927052200390793911046812, 8.171469370219044666060398403866, 8.841976384039721618031774349782, 9.409927696023136640060827762838

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