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Properties

Label 1-287-287.282-r0-0-0
Degree $1$
Conductor $287$
Sign $0.914 - 0.404i$
Analytic cond. $1.33282$
Root an. cond. $1.33282$
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  + (0.978 − 0.207i)2-s + (0.866 + 0.5i)3-s + (0.913 − 0.406i)4-s + (0.104 − 0.994i)5-s + (0.951 + 0.309i)6-s + (0.809 − 0.587i)8-s + (0.5 + 0.866i)9-s + (−0.104 − 0.994i)10-s + (−0.994 + 0.104i)11-s + (0.994 + 0.104i)12-s + (−0.951 − 0.309i)13-s + (0.587 − 0.809i)15-s + (0.669 − 0.743i)16-s + (0.994 − 0.104i)17-s + (0.669 + 0.743i)18-s + (0.743 + 0.669i)19-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)2-s + (0.866 + 0.5i)3-s + (0.913 − 0.406i)4-s + (0.104 − 0.994i)5-s + (0.951 + 0.309i)6-s + (0.809 − 0.587i)8-s + (0.5 + 0.866i)9-s + (−0.104 − 0.994i)10-s + (−0.994 + 0.104i)11-s + (0.994 + 0.104i)12-s + (−0.951 − 0.309i)13-s + (0.587 − 0.809i)15-s + (0.669 − 0.743i)16-s + (0.994 − 0.104i)17-s + (0.669 + 0.743i)18-s + (0.743 + 0.669i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.914 - 0.404i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (282, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.914 - 0.404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.826163187 - 0.5969694630i\)
\(L(\frac12)\) \(\approx\) \(2.826163187 - 0.5969694630i\)
\(L(1)\) \(\approx\) \(2.261077108 - 0.3028644306i\)
\(L(1)\) \(\approx\) \(2.261077108 - 0.3028644306i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.978 - 0.207i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.104 - 0.994i)T \)
11 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
17 \( 1 + (0.994 - 0.104i)T \)
19 \( 1 + (0.743 + 0.669i)T \)
23 \( 1 + (-0.978 + 0.207i)T \)
29 \( 1 + (-0.587 + 0.809i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.309 + 0.951i)T \)
47 \( 1 + (-0.207 - 0.978i)T \)
53 \( 1 + (-0.406 - 0.913i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (0.406 + 0.913i)T \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.743 - 0.669i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−25.561340221350548144005543517044, −24.57742487407348845047604935717, −23.825905629255279609502922620554, −23.00302553799654680276155016579, −21.91931334241858780189367836233, −21.245933248003707973154805712010, −20.24315825823051643597098597337, −19.33681736348669015128913272898, −18.46184567442406405568293226658, −17.42457789223705466368118767139, −15.96801296054379028578197819476, −15.201307151992873017083532136257, −14.25595832252517487716451810213, −13.85703711354859226151873585839, −12.71205942656196382755481968420, −11.88806419648972463617556684667, −10.64115115561381232745101726581, −9.60488301937690274760152869672, −7.883324010188719413242551073839, −7.42520748023360286694823117666, −6.39936719925363964080035264290, −5.22297013256674510899195200415, −3.74207462862302667739256855105, −2.83441808986625433956922436140, −2.05431779178180691512987905193, 1.64646403383423894731021058738, 2.799365034455296155077805284841, 3.87132037041735592119846515458, 4.99768115535640771799645937989, 5.56669367144506473463637131598, 7.4770621845879555045575493938, 8.13506135903944346917765864277, 9.74879070419712192301431596016, 10.13812527867191033716221998236, 11.701432429139469079536115063058, 12.66835474292779501717684937181, 13.38390640786870740496656038062, 14.32917714410183727686563505953, 15.17024051879946308783826410649, 16.123179337867392115276242401, 16.718066413014760459123756210432, 18.433499613093662894288559185724, 19.59407771343952624084824023215, 20.35043114616669870643625474803, 20.86686429190709790160731127838, 21.69012666553430625050971294490, 22.60891584756666320264504414768, 23.84479291110165364605481614489, 24.45722163179279047423606272109, 25.32328448955276217634684331740

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