Properties

Label 1-287-287.163-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.895 + 0.444i$
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 + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 6-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s − 13-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 6-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s − 13-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1702210652 - 0.7266291057i\)
\(L(\frac12)\) \(\approx\) \(-0.1702210652 - 0.7266291057i\)
\(L(1)\) \(\approx\) \(0.4421165164 - 0.6346373332i\)
\(L(1)\) \(\approx\) \(0.4421165164 - 0.6346373332i\)

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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 - T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 - T \)
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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

−26.10141654245714274310678562404, −25.50699493709654004008715709319, −24.37905800166433459314377747705, −23.40527497528516643654135013940, −22.314941205057051069142221230966, −21.97313817965315550267805866797, −20.2903925281931775734986395190, −19.60003760151349910788622574198, −18.85820571304383161737647334665, −17.60006277780616132635591918853, −16.88229548155143231098732181760, −15.72520897511560000589209716472, −14.9953465626092827998167310007, −14.608434620675630181694332395539, −13.50776485906500107145259460239, −11.8189357964073549095251294548, −10.64393438519382269676658750536, −9.848089438931456407955003419069, −9.09334741276895573125701755779, −7.73751993912294699583849480277, −7.24514437448231974853072751780, −5.82293199868346826532629283113, −4.626735635443396431107394867481, −3.64734935570412803259013365446, −2.12098991840302151426405260034, 0.557383863455694596939549068794, 1.717079086978588353697225836421, 3.023402639086240594328495589, 4.02553974222150442813113151613, 5.48858875071133231703087383981, 7.232576431466960809559922166522, 7.96374830575090157162848891309, 8.8928154167132688897873119935, 9.60745811765579784143539551696, 11.131938035248249944476854187421, 12.24163263848098703811352653315, 12.408818176135697943063547696274, 13.71940240077945037467519051389, 14.45226180676236636454591977978, 16.203036868724470589000722294877, 16.879080298834496152738252063542, 17.96830234230203987519540930471, 18.90836720646293971023849727798, 19.51569314057264214655060443662, 20.33086416036752064304921214422, 20.961447556119075990096407226399, 22.24007991419325019170895915323, 23.180746915921119053622813269970, 24.45029796177907890665626517015, 24.79778763885177023572600301819

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