Properties

Label 1-287-287.9-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.992 + 0.122i$
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  + (0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s i·6-s − 8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s + (0.866 − 0.5i)12-s + i·13-s i·15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.5 + 0.866i)18-s + (−0.866 + 0.5i)19-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s i·6-s − 8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s + (0.866 − 0.5i)12-s + i·13-s i·15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.5 + 0.866i)18-s + (−0.866 + 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.992 + 0.122i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.992 + 0.122i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05026865222 + 0.8173242680i\)
\(L(\frac12)\) \(\approx\) \(0.05026865222 + 0.8173242680i\)
\(L(1)\) \(\approx\) \(0.6712907875 + 0.5778309764i\)
\(L(1)\) \(\approx\) \(0.6712907875 + 0.5778309764i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + iT \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + iT \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + iT \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 - iT \)
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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

−24.8919142962686429995666696652, −23.74756872400597952006738414790, −23.25743858027309765939540396327, −22.26337510819700874784450665657, −21.403713460044907064822548109556, −20.75688065844549038135745090441, −20.079030931305742327206968305810, −18.66373988987038071768339585079, −17.770039625181748360607089127657, −17.030362113289352050542887920414, −15.74231446560671269759110114700, −15.06131024779695266730242304537, −13.57197784496004423913242864350, −12.81681819320423663757615623324, −12.0865946918288754374437449028, −11.04373845841298590048394415127, −9.99822307103379233990587981032, −9.58245570532143038046813353216, −8.068676359831567463994426507893, −6.23476407485783040766635395643, −5.28716828052083519550529269262, −4.78600175969462718733791984722, −3.470777061621600546035663163168, −1.969237866570558246460071344928, −0.49819433214172768114379279783, 2.01824382114931415695349677823, 3.47685523842693039262802585815, 4.90471775061877795884406752159, 5.892497323748532339801276129, 6.5611019043723530858873172043, 7.46039815327973351443935415070, 8.54434308692807511862811608299, 10.1376677792015295072189765380, 11.02112389051435034212863961089, 12.23076941431639836894891393883, 13.02136487674105328655505932695, 14.05545379472637332912104610898, 14.704361488085813979319702251628, 16.12535684759570065636518674391, 16.64961744188467261654295826414, 17.71071147738841877765410338089, 18.4452128483249781780037895636, 19.09655526061407572105569848149, 21.1826600563096765986673555812, 21.6238499317409192538761245960, 22.546128084004096355914452697442, 23.44727980321577759049927652485, 23.88200283000278672346612742166, 24.99217330756089230612364212238, 25.842861918486224783506750334450

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