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Properties

Label 1-287-287.251-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.389 + 0.920i$
Analytic cond. $30.8424$
Root an. cond. $30.8424$
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  + (−0.309 + 0.951i)2-s − i·3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (0.951 + 0.309i)6-s + (0.809 − 0.587i)8-s − 9-s + (0.809 − 0.587i)10-s + (−0.587 − 0.809i)11-s + (−0.587 + 0.809i)12-s + (−0.951 − 0.309i)13-s + (−0.587 + 0.809i)15-s + (0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s + (0.309 − 0.951i)18-s + (−0.951 + 0.309i)19-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s − i·3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (0.951 + 0.309i)6-s + (0.809 − 0.587i)8-s − 9-s + (0.809 − 0.587i)10-s + (−0.587 − 0.809i)11-s + (−0.587 + 0.809i)12-s + (−0.951 − 0.309i)13-s + (−0.587 + 0.809i)15-s + (0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s + (0.309 − 0.951i)18-s + (−0.951 + 0.309i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.389 + 0.920i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.389 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02066773276 + 0.03119814572i\)
\(L(\frac12)\) \(\approx\) \(0.02066773276 + 0.03119814572i\)
\(L(1)\) \(\approx\) \(0.5282765150 - 0.1096462889i\)
\(L(1)\) \(\approx\) \(0.5282765150 - 0.1096462889i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 - iT \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (-0.951 + 0.309i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.587 - 0.809i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.309 + 0.951i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.587 + 0.809i)T \)
71 \( 1 + (-0.587 - 0.809i)T \)
73 \( 1 + T \)
79 \( 1 + iT \)
83 \( 1 - T \)
89 \( 1 + (0.951 - 0.309i)T \)
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

−26.2618288371138569108278464005, −25.54961804520272489362543645191, −23.64165565409548231714646403766, −23.03664031661121009105192034401, −21.96588791567275962474291493270, −21.55307649921809511523385480853, −20.369536081133056384568720761753, −19.686539712772585211877664663098, −18.9661489495003411027809453310, −17.64763003688306205866918181235, −17.02900288614142827186272113869, −15.64202368798136138747286051610, −15.01048149879606891316574612592, −13.92991114076731747577077184860, −12.53663915937768998193972532309, −11.745184089856135070260853138504, −10.66539958151823095189533793057, −10.25564999934999370812723826972, −9.06188210486932779510738496456, −8.138241777277962834325095149313, −6.939774611981438685836135388985, −5.00322201293056704211648334289, −4.27010668845106669949050576465, −3.22211247068561373552604307447, −2.19426289228481967050272913474, 0.018550613062762027050536024969, 0.76587859352790840308647731221, 2.59473790514954902161892399996, 4.37198551606160517797497939082, 5.40752614971410083779084095659, 6.52355112591131929073065131536, 7.51445403985258711971261054104, 8.24166921052364505074934305975, 8.960661453641624206648728229813, 10.48082081193990480392659763850, 11.730035793131516981402409593881, 12.76747130206111533800589152523, 13.51751543130544333997876402834, 14.591972237485250752851005908647, 15.549751663850667546423227134186, 16.53408295070467286861380541658, 17.25670464153243512483762778530, 18.27276853854396189115327217184, 19.16211719540853663266890891209, 19.6562925651200838007481976496, 20.92645121702869329314724931416, 22.55962473224938814245343922434, 23.1084981196675410394782849823, 24.20209087445321641251444855063, 24.43370746381753036295774193246

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