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Properties

Label 1-287-287.200-r0-0-0
Degree $1$
Conductor $287$
Sign $0.875 + 0.483i$
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  + (−0.669 − 0.743i)2-s + (−0.866 + 0.5i)3-s + (−0.104 + 0.994i)4-s + (−0.913 + 0.406i)5-s + (0.951 + 0.309i)6-s + (0.809 − 0.587i)8-s + (0.5 − 0.866i)9-s + (0.913 + 0.406i)10-s + (0.406 − 0.913i)11-s + (−0.406 − 0.913i)12-s + (−0.951 − 0.309i)13-s + (0.587 − 0.809i)15-s + (−0.978 − 0.207i)16-s + (−0.406 + 0.913i)17-s + (−0.978 + 0.207i)18-s + (0.207 − 0.978i)19-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (−0.866 + 0.5i)3-s + (−0.104 + 0.994i)4-s + (−0.913 + 0.406i)5-s + (0.951 + 0.309i)6-s + (0.809 − 0.587i)8-s + (0.5 − 0.866i)9-s + (0.913 + 0.406i)10-s + (0.406 − 0.913i)11-s + (−0.406 − 0.913i)12-s + (−0.951 − 0.309i)13-s + (0.587 − 0.809i)15-s + (−0.978 − 0.207i)16-s + (−0.406 + 0.913i)17-s + (−0.978 + 0.207i)18-s + (0.207 − 0.978i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.875 + 0.483i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.875 + 0.483i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4272073082 + 0.1100191030i\)
\(L(\frac12)\) \(\approx\) \(0.4272073082 + 0.1100191030i\)
\(L(1)\) \(\approx\) \(0.4919064738 + 0.02052071238i\)
\(L(1)\) \(\approx\) \(0.4919064738 + 0.02052071238i\)

Euler product

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

−25.04114705527845252982376296527, −24.675318782759148832251706494763, −23.79072528706837235172579604670, −22.85721854086573204838247146049, −22.49650470057942771576409385793, −20.64134289983135899556004646293, −19.69947082714749108552396437342, −18.89073166152226904453579650055, −18.10925598492073859613654865393, −16.96671055699219892831734425639, −16.63926881777060222700374425481, −15.514075301569357598978488615968, −14.696575778813935295605181642935, −13.35970550767731665922826161029, −12.13002668061636959667012740681, −11.54184530117756264913508550638, −10.27006828940723307780304566377, −9.30237451772771894166241800819, −7.974254108071669303717998736376, −7.281793468888327636861210191703, −6.46456772407502950487570090025, −5.0823169784742831168208215865, −4.42619376631716601620392557134, −2.0283560589886853393635812059, −0.57362983513704162704679810452, 0.93323860442060285937562103867, 2.90686699486861939227663196375, 3.85258020885399820770487087326, 4.912258522350141677383515210538, 6.52026301666800568276066164535, 7.50362712190583900978224349199, 8.699176148724963198611961275964, 9.71255476660379152809949584837, 10.85776721510785910388695036892, 11.27362363923555089053339806836, 12.1558587302213879580961620282, 13.12098414318042630403444884887, 14.774954603503744668873455211542, 15.724820740933405651738224445923, 16.63118949531348315688390500225, 17.419140681339141363474102661435, 18.29426120145784298069502532462, 19.43247689396398032590838350617, 19.829573835276694029759472531018, 21.27170011172413657565625754630, 21.923211830956087330967185226374, 22.58352147531685139330111967922, 23.65573633634397545979879943336, 24.60867125287079660794699191933, 26.18121635978963643822451031770

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