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Properties

Label 1-283-283.55-r1-0-0
Degree $1$
Conductor $283$
Sign $0.0210 - 0.999i$
Analytic cond. $30.4125$
Root an. cond. $30.4125$
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  + (0.892 + 0.451i)2-s + (−0.662 + 0.749i)3-s + (0.593 + 0.805i)4-s + (0.798 + 0.602i)5-s + (−0.929 + 0.369i)6-s + (−0.970 − 0.242i)7-s + (0.166 + 0.986i)8-s + (−0.122 − 0.992i)9-s + (0.441 + 0.897i)10-s + (−0.993 − 0.111i)11-s + (−0.996 − 0.0890i)12-s + (0.144 − 0.989i)13-s + (−0.756 − 0.654i)14-s + (−0.979 + 0.199i)15-s + (−0.296 + 0.955i)16-s + (−0.188 − 0.982i)17-s + ⋯
L(s)  = 1  + (0.892 + 0.451i)2-s + (−0.662 + 0.749i)3-s + (0.593 + 0.805i)4-s + (0.798 + 0.602i)5-s + (−0.929 + 0.369i)6-s + (−0.970 − 0.242i)7-s + (0.166 + 0.986i)8-s + (−0.122 − 0.992i)9-s + (0.441 + 0.897i)10-s + (−0.993 − 0.111i)11-s + (−0.996 − 0.0890i)12-s + (0.144 − 0.989i)13-s + (−0.756 − 0.654i)14-s + (−0.979 + 0.199i)15-s + (−0.296 + 0.955i)16-s + (−0.188 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0210 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0210 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.0210 - 0.999i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.0210 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05163605922 - 0.05055940363i\)
\(L(\frac12)\) \(\approx\) \(0.05163605922 - 0.05055940363i\)
\(L(1)\) \(\approx\) \(0.9306551237 + 0.5467029620i\)
\(L(1)\) \(\approx\) \(0.9306551237 + 0.5467029620i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.892 + 0.451i)T \)
3 \( 1 + (-0.662 + 0.749i)T \)
5 \( 1 + (0.798 + 0.602i)T \)
7 \( 1 + (-0.970 - 0.242i)T \)
11 \( 1 + (-0.993 - 0.111i)T \)
13 \( 1 + (0.144 - 0.989i)T \)
17 \( 1 + (-0.188 - 0.982i)T \)
19 \( 1 + (-0.784 - 0.619i)T \)
23 \( 1 + (-0.987 - 0.155i)T \)
29 \( 1 + (-0.645 + 0.763i)T \)
31 \( 1 + (0.871 - 0.490i)T \)
37 \( 1 + (-0.951 - 0.306i)T \)
41 \( 1 + (-0.770 - 0.637i)T \)
43 \( 1 + (-0.100 + 0.994i)T \)
47 \( 1 + (0.929 + 0.369i)T \)
53 \( 1 + (0.296 + 0.955i)T \)
59 \( 1 + (-0.848 + 0.528i)T \)
61 \( 1 + (-0.997 + 0.0667i)T \)
67 \( 1 + (0.420 - 0.907i)T \)
71 \( 1 + (0.593 - 0.805i)T \)
73 \( 1 + (-0.871 - 0.490i)T \)
79 \( 1 + (-0.100 - 0.994i)T \)
83 \( 1 + (0.902 - 0.431i)T \)
89 \( 1 + (-0.911 + 0.410i)T \)
97 \( 1 + (0.628 - 0.777i)T \)
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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

−25.35219555126922483470392859372, −24.37012501682198784379479575281, −23.70136466066899217479429732827, −22.94848374847204770559800251272, −21.88946840294223776426870858679, −21.354486213868142519241716987, −20.23234859663116725428735155251, −19.12499928073304003387321679244, −18.58171540621190782808787366206, −17.22933671318985764791841288075, −16.386687125059743287050613151290, −15.46816058614964753919920116723, −13.93550813855027811790983420674, −13.31426835680227704513431265815, −12.58116580429364167386311746292, −11.93105724257680981709529558790, −10.55500809524868614184784978068, −9.89285761957908858863315710638, −8.38638334363348769469892252643, −6.76622252244460396334556790052, −6.07882490954178334054177068471, −5.31857342959255645605213932469, −4.066004814387087432511996004891, −2.385998432649248162770639482168, −1.64204174928566615658485000371, 0.01564051326562048932408669425, 2.61350524563028252326934186668, 3.40345738054085225146637566460, 4.7657253171221746250816109702, 5.73294447756602027447730653118, 6.39371610948107968027971760594, 7.461343819220721800366288756057, 9.099791885728984217997845715222, 10.34937007294344879987251104910, 10.81845497867390873848809607978, 12.20673806836887482256848446070, 13.1596220547614541175855981107, 13.9089726961821258291173568618, 15.22452441465883546362493705394, 15.71582050217598031341292705731, 16.673512896610063347725569914693, 17.555795082078814238236245786474, 18.42242373374317629174016944515, 20.11662833670991210139023082952, 20.92217684557317353374773673565, 21.7990077395965726388168869897, 22.526452668747671302891853514179, 22.9839359689501143775283431485, 24.00463092849141082532307455323, 25.231923739302164936241704140886

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