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Properties

Label 1-4021-4021.28-r1-0-0
Degree $1$
Conductor $4021$
Sign $0.992 + 0.126i$
Analytic cond. $432.116$
Root an. cond. $432.116$
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  + (0.585 + 0.810i)2-s + (0.862 + 0.506i)3-s + (−0.314 + 0.949i)4-s + (0.640 − 0.767i)5-s + (0.0936 + 0.995i)6-s − i·7-s + (−0.953 + 0.300i)8-s + (0.486 + 0.873i)9-s + (0.997 + 0.0702i)10-s + (0.437 − 0.899i)11-s + (−0.752 + 0.658i)12-s + (0.972 + 0.232i)13-s + (0.810 − 0.585i)14-s + (0.941 − 0.337i)15-s + (−0.801 − 0.597i)16-s + (−0.845 + 0.533i)17-s + ⋯
L(s)  = 1  + (0.585 + 0.810i)2-s + (0.862 + 0.506i)3-s + (−0.314 + 0.949i)4-s + (0.640 − 0.767i)5-s + (0.0936 + 0.995i)6-s − i·7-s + (−0.953 + 0.300i)8-s + (0.486 + 0.873i)9-s + (0.997 + 0.0702i)10-s + (0.437 − 0.899i)11-s + (−0.752 + 0.658i)12-s + (0.972 + 0.232i)13-s + (0.810 − 0.585i)14-s + (0.941 − 0.337i)15-s + (−0.801 − 0.597i)16-s + (−0.845 + 0.533i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4021\)
Sign: $0.992 + 0.126i$
Analytic conductor: \(432.116\)
Root analytic conductor: \(432.116\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4021} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4021,\ (1:\ ),\ 0.992 + 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.590982223 + 0.3537090072i\)
\(L(\frac12)\) \(\approx\) \(5.590982223 + 0.3537090072i\)
\(L(1)\) \(\approx\) \(2.079627469 + 0.6870952989i\)
\(L(1)\) \(\approx\) \(2.079627469 + 0.6870952989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4021 \( 1 \)
good2 \( 1 + (0.585 + 0.810i)T \)
3 \( 1 + (0.862 + 0.506i)T \)
5 \( 1 + (0.640 - 0.767i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.437 - 0.899i)T \)
13 \( 1 + (0.972 + 0.232i)T \)
17 \( 1 + (-0.845 + 0.533i)T \)
19 \( 1 + (0.155 - 0.987i)T \)
23 \( 1 + (0.479 - 0.877i)T \)
29 \( 1 + (0.247 + 0.968i)T \)
31 \( 1 + (0.726 + 0.687i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.915 - 0.402i)T \)
43 \( 1 + (-0.109 + 0.994i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.458 - 0.888i)T \)
59 \( 1 + (0.964 - 0.262i)T \)
61 \( 1 + (0.329 - 0.944i)T \)
67 \( 1 + (-0.0234 + 0.999i)T \)
71 \( 1 + (-0.472 + 0.881i)T \)
73 \( 1 + (0.946 + 0.322i)T \)
79 \( 1 + (0.962 - 0.270i)T \)
83 \( 1 + (-0.992 + 0.124i)T \)
89 \( 1 + (0.899 + 0.437i)T \)
97 \( 1 + (-0.373 - 0.927i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.361172815000258034649716334694, −18.04118307796373583831670070034, −17.33297356194067634345589682159, −15.75704668900745850146479684493, −15.22478338011903177762074012546, −14.84924250417748961428702064968, −14.01027959857455651524541771851, −13.47951789851804642827599584910, −13.02400362389285618283740665821, −12.02544733503743636092141091632, −11.70764770571875134412932227619, −10.75369943316000765183703929335, −9.83362661313542535679715229924, −9.47551993547374001682668916342, −8.77527494477076261052443762723, −7.883054704950167443873020605605, −6.82559259949670546978891952873, −6.271069111903734021420723365824, −5.64959822477936121820362608261, −4.60871737140726833188871601760, −3.68210364104686685941788213895, −3.064145699096806842396632736215, −2.275652408969061437038405259416, −1.86402545824473904996623322862, −1.01384019547217884421442884032, 0.56516011803037045558000823348, 1.51528530442567036995172444384, 2.66842112368636379221377115088, 3.45833285779076179641980230544, 4.175707271880576123247925683565, 4.686440274474063946454004287681, 5.45304106972356538508321735769, 6.53635154969067197433330301814, 6.834075463756801278295983229626, 8.05251430952542481605170047219, 8.59801722753496717951288303686, 8.95890139710851268412643393501, 9.77023717527212046545588642470, 10.82644968709504415100074121659, 11.26903978808186889150625133584, 12.74957770732309065685848373499, 13.022978564914837654088527409069, 13.76108373682776201127961352173, 14.13052595988315082872967989969, 14.7137859325505687575250141229, 15.927863027838317309225496218657, 16.04319994190992259314197526735, 16.690883030021520163721934970516, 17.50645488191074126311745738701, 17.95717985866492368120566822853

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