↑

Properties

Label 1-287-287.181-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.525 - 0.850i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.707 − 0.707i)3-s + (0.809 − 0.587i)4-s + (−0.587 − 0.809i)5-s + (−0.453 + 0.891i)6-s + (−0.587 + 0.809i)8-s − i·9-s + (0.809 + 0.587i)10-s + (−0.987 + 0.156i)11-s + (0.156 − 0.987i)12-s + (0.891 + 0.453i)13-s + (−0.987 − 0.156i)15-s + (0.309 − 0.951i)16-s + (−0.156 − 0.987i)17-s + (0.309 + 0.951i)18-s + (0.891 − 0.453i)19-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.707 − 0.707i)3-s + (0.809 − 0.587i)4-s + (−0.587 − 0.809i)5-s + (−0.453 + 0.891i)6-s + (−0.587 + 0.809i)8-s − i·9-s + (0.809 + 0.587i)10-s + (−0.987 + 0.156i)11-s + (0.156 − 0.987i)12-s + (0.891 + 0.453i)13-s + (−0.987 − 0.156i)15-s + (0.309 − 0.951i)16-s + (−0.156 − 0.987i)17-s + (0.309 + 0.951i)18-s + (0.891 − 0.453i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3585805969 - 0.6427876045i\)
\(L(\frac12)\) \(\approx\) \(0.3585805969 - 0.6427876045i\)
\(L(1)\) \(\approx\) \(0.6709288113 - 0.3127360233i\)
\(L(1)\) \(\approx\) \(0.6709288113 - 0.3127360233i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (-0.987 + 0.156i)T \)
13 \( 1 + (0.891 + 0.453i)T \)
17 \( 1 + (-0.156 - 0.987i)T \)
19 \( 1 + (0.891 - 0.453i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (0.156 - 0.987i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (-0.453 + 0.891i)T \)
53 \( 1 + (-0.156 + 0.987i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (-0.951 - 0.309i)T \)
67 \( 1 + (0.987 + 0.156i)T \)
71 \( 1 + (0.987 - 0.156i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.453 - 0.891i)T \)
97 \( 1 + (0.987 + 0.156i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.946069859902724315777062900102, −25.5466446033145204442985250288, −24.20400496506694313585498415395, −23.052756754231319051433056637611, −21.85004200802987131529572132310, −21.21146213723083866190612705927, −20.15548845012117435977481123916, −19.60206670284370856802163776355, −18.54928332648448863360371625337, −17.975249324283897154271077273527, −16.47624895746940909381638316888, −15.72349388550011502030544496852, −15.16455023400224899912674266241, −13.91661279464724397556134938834, −12.70988282785927560001304198275, −11.32134479688964973911181660846, −10.615115824273212339932581213, −9.97765430794503515876336982813, −8.639801634219542034299709105870, −8.01185796051140306133179583483, −7.088302470442140295196625960564, −5.54063597708867173778807215323, −3.63702875011099051611333233418, −3.238959757501564773982301736625, −1.85618065607998625439909005966, 0.60174237183551259951742711910, 1.902628635300361341863878891653, 3.160723129338229122832753774730, 4.846323199683465090914604665914, 6.23319279391482175088149280725, 7.38793905074382068090074790414, 8.04873243309760030592589351465, 8.89153398586389677539012307015, 9.70504653921296838914599180639, 11.19314173792180313040479095278, 12.04243300834558549610115925000, 13.18137581220035833593039388851, 14.11500242947787825086181563743, 15.47925889887186779981526765576, 15.91470763935660876580606897036, 17.03442964226273229084143226006, 18.29969377825564953461841272308, 18.59706715394404048982740035823, 19.75608371320901892463417279091, 20.48157964463060130038230791489, 20.94249260479403986780600877824, 22.9843226879412440384657405056, 23.84670269318537546906603825783, 24.42358521293151452699682690794, 25.21808736462385012302125236216

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