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Properties

Label 1-283-283.51-r0-0-0
Degree $1$
Conductor $283$
Sign $0.885 - 0.464i$
Analytic cond. $1.31424$
Root an. cond. $1.31424$
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  + (0.480 + 0.876i)2-s + (0.231 − 0.972i)3-s + (−0.538 + 0.842i)4-s + (−0.892 + 0.451i)5-s + (0.964 − 0.264i)6-s + (0.593 − 0.805i)7-s + (−0.997 − 0.0667i)8-s + (−0.892 − 0.451i)9-s + (−0.824 − 0.565i)10-s + (−0.538 − 0.842i)11-s + (0.695 + 0.718i)12-s + (0.964 − 0.264i)13-s + (0.991 + 0.133i)14-s + (0.231 + 0.972i)15-s + (−0.420 − 0.907i)16-s + (0.991 − 0.133i)17-s + ⋯
L(s)  = 1  + (0.480 + 0.876i)2-s + (0.231 − 0.972i)3-s + (−0.538 + 0.842i)4-s + (−0.892 + 0.451i)5-s + (0.964 − 0.264i)6-s + (0.593 − 0.805i)7-s + (−0.997 − 0.0667i)8-s + (−0.892 − 0.451i)9-s + (−0.824 − 0.565i)10-s + (−0.538 − 0.842i)11-s + (0.695 + 0.718i)12-s + (0.964 − 0.264i)13-s + (0.991 + 0.133i)14-s + (0.231 + 0.972i)15-s + (−0.420 − 0.907i)16-s + (0.991 − 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.885 - 0.464i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ 0.885 - 0.464i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.283779535 - 0.3162818864i\)
\(L(\frac12)\) \(\approx\) \(1.283779535 - 0.3162818864i\)
\(L(1)\) \(\approx\) \(1.191778579 + 0.02710696261i\)
\(L(1)\) \(\approx\) \(1.191778579 + 0.02710696261i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.480 + 0.876i)T \)
3 \( 1 + (0.231 - 0.972i)T \)
5 \( 1 + (-0.892 + 0.451i)T \)
7 \( 1 + (0.593 - 0.805i)T \)
11 \( 1 + (-0.538 - 0.842i)T \)
13 \( 1 + (0.964 - 0.264i)T \)
17 \( 1 + (0.991 - 0.133i)T \)
19 \( 1 + (0.964 - 0.264i)T \)
23 \( 1 + (-0.166 - 0.986i)T \)
29 \( 1 + (-0.0334 + 0.999i)T \)
31 \( 1 + (0.100 - 0.994i)T \)
37 \( 1 + (-0.944 + 0.328i)T \)
41 \( 1 + (-0.997 + 0.0667i)T \)
43 \( 1 + (0.784 - 0.619i)T \)
47 \( 1 + (0.964 + 0.264i)T \)
53 \( 1 + (-0.420 + 0.907i)T \)
59 \( 1 + (-0.296 - 0.955i)T \)
61 \( 1 + (-0.824 + 0.565i)T \)
67 \( 1 + (0.695 - 0.718i)T \)
71 \( 1 + (-0.538 - 0.842i)T \)
73 \( 1 + (0.100 + 0.994i)T \)
79 \( 1 + (0.784 + 0.619i)T \)
83 \( 1 + (-0.645 + 0.763i)T \)
89 \( 1 + (0.784 - 0.619i)T \)
97 \( 1 + (-0.166 - 0.986i)T \)
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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

−25.738018484968259623427009200481, −24.61791334264110821364862055814, −23.42202240479837470662541385251, −22.9487843795555949395477014917, −21.81813681795684475351137896811, −20.86854554892393806732200926428, −20.63715856644732197178435130554, −19.517012090983214953373556308819, −18.648137327823150926890540193394, −17.551794685148540402029952750, −15.97835174435764978524190101623, −15.4783404408166145150984884776, −14.59594043044155225399956130821, −13.59394483681800684286262998580, −12.21687541260126183342050293259, −11.69856388150689134916761630067, −10.71159249849711884775176856387, −9.66271200829783915140704153014, −8.76357207038685778444210639296, −7.81562539726295284714624890236, −5.63551496096070027060475511575, −4.98142762483028274030333826286, −3.94857204361016425590887947236, −3.07753307205154230071603367000, −1.601627921784642349606049376601, 0.8131898631049679681316619419, 2.997851832768510554572660737286, 3.75244684197849828377189894897, 5.2349628414163808550676809626, 6.36611344358344327202235263765, 7.40108609766215889998541305023, 7.93156079045537433815947943309, 8.72729074467878280163457905654, 10.70686641868919312912713783877, 11.67212666054118124128514499958, 12.61988033353924687320296246983, 13.81172338381231307397549720063, 14.11333957772556761840680747566, 15.27967986807702077196176520964, 16.24512839887378752964060532053, 17.16633996633264666852376792012, 18.4737417410987473907893306537, 18.60239270398254021856590350875, 20.18332886553689484088522197550, 20.86037653773212048322869320291, 22.361938616915253123044152432343, 23.1223999259322704874233765131, 23.90850743229721495216456000882, 24.19764495847606610225296327236, 25.47143418398417490159176511137

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