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Properties

Label 1-287-287.60-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.901 - 0.432i$
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  + (0.406 + 0.913i)2-s + (0.965 + 0.258i)3-s + (−0.669 + 0.743i)4-s + (−0.207 + 0.978i)5-s + (0.156 + 0.987i)6-s + (−0.951 − 0.309i)8-s + (0.866 + 0.5i)9-s + (−0.978 + 0.207i)10-s + (−0.544 + 0.838i)11-s + (−0.838 + 0.544i)12-s + (0.987 − 0.156i)13-s + (−0.453 + 0.891i)15-s + (−0.104 − 0.994i)16-s + (0.838 + 0.544i)17-s + (−0.104 + 0.994i)18-s + (−0.629 + 0.777i)19-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (0.965 + 0.258i)3-s + (−0.669 + 0.743i)4-s + (−0.207 + 0.978i)5-s + (0.156 + 0.987i)6-s + (−0.951 − 0.309i)8-s + (0.866 + 0.5i)9-s + (−0.978 + 0.207i)10-s + (−0.544 + 0.838i)11-s + (−0.838 + 0.544i)12-s + (0.987 − 0.156i)13-s + (−0.453 + 0.891i)15-s + (−0.104 − 0.994i)16-s + (0.838 + 0.544i)17-s + (−0.104 + 0.994i)18-s + (−0.629 + 0.777i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.901 - 0.432i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.901 - 0.432i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5524839338 + 2.429479881i\)
\(L(\frac12)\) \(\approx\) \(-0.5524839338 + 2.429479881i\)
\(L(1)\) \(\approx\) \(0.8763614282 + 1.265498828i\)
\(L(1)\) \(\approx\) \(0.8763614282 + 1.265498828i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.406 + 0.913i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (-0.207 + 0.978i)T \)
11 \( 1 + (-0.544 + 0.838i)T \)
13 \( 1 + (0.987 - 0.156i)T \)
17 \( 1 + (0.838 + 0.544i)T \)
19 \( 1 + (-0.629 + 0.777i)T \)
23 \( 1 + (-0.913 + 0.406i)T \)
29 \( 1 + (-0.891 - 0.453i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (-0.933 - 0.358i)T \)
53 \( 1 + (-0.0523 - 0.998i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (0.994 - 0.104i)T \)
67 \( 1 + (-0.998 + 0.0523i)T \)
71 \( 1 + (0.453 + 0.891i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.258 + 0.965i)T \)
83 \( 1 + T \)
89 \( 1 + (0.777 + 0.629i)T \)
97 \( 1 + (-0.453 + 0.891i)T \)
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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

−24.47732779182837834300515397143, −23.95035554521335082011686103937, −23.11224160663092067839743352118, −21.67957181427428702999197028485, −20.8997906168539902766568445332, −20.484012203165312528526974378924, −19.420911971256033559492013605470, −18.82115792817416195636041068612, −17.848318724676212798266786081736, −16.28283503778949181511846250288, −15.45866107704934751218170656854, −14.1998870709080167055531691188, −13.499556939048708282059508946684, −12.80486389074799479003832479297, −11.86709669140715539922122412281, −10.72380420826417538729141304065, −9.52772942981937484739972175236, −8.70082767572276629582474468065, −7.98474272197862593500498818972, −6.21835491310359953280707603787, −4.968899577732436170417220577767, −3.89086519036866319638522410421, −2.96031080769922357193052208677, −1.65058700597504090316028254460, −0.57656966994279216125294390921, 2.13746975139782260647945537339, 3.484758876482668178371442098977, 4.05497720550758642292203257611, 5.58433062261066669812173742896, 6.71771048333066201111416443707, 7.780334988664385744020789810069, 8.27929161718277019394032361370, 9.70542867788072426282939336374, 10.50246608786241380014890118247, 12.10234997349404264131667758264, 13.19380267896685285177410544645, 14.02451324491211393485760639898, 14.87194215140210325045691314477, 15.41644013736020619707966361463, 16.27995234637511991181674909753, 17.6313568865585067567823345826, 18.52891656570820902408550925216, 19.25146613904479010676683474928, 20.72843175099634714002403035859, 21.25775181000955438052126235336, 22.43413653551570537129928783326, 23.124967355811417417950190043324, 24.0126972778289809868960098067, 25.20999410110197640062047579555, 25.94223566035854350601460499240

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