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Properties

Label 1-287-287.116-r1-0-0
Degree $1$
Conductor $287$
Sign $0.376 + 0.926i$
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 + (0.258 − 0.965i)3-s + (−0.669 + 0.743i)4-s + (0.207 − 0.978i)5-s + (−0.987 + 0.156i)6-s + (0.951 + 0.309i)8-s + (−0.866 − 0.5i)9-s + (−0.978 + 0.207i)10-s + (0.838 + 0.544i)11-s + (0.544 + 0.838i)12-s + (−0.156 − 0.987i)13-s + (−0.891 − 0.453i)15-s + (−0.104 − 0.994i)16-s + (−0.544 + 0.838i)17-s + (−0.104 + 0.994i)18-s + (−0.777 − 0.629i)19-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (0.258 − 0.965i)3-s + (−0.669 + 0.743i)4-s + (0.207 − 0.978i)5-s + (−0.987 + 0.156i)6-s + (0.951 + 0.309i)8-s + (−0.866 − 0.5i)9-s + (−0.978 + 0.207i)10-s + (0.838 + 0.544i)11-s + (0.544 + 0.838i)12-s + (−0.156 − 0.987i)13-s + (−0.891 − 0.453i)15-s + (−0.104 − 0.994i)16-s + (−0.544 + 0.838i)17-s + (−0.104 + 0.994i)18-s + (−0.777 − 0.629i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.376 + 0.926i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.376 + 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3012752972 - 0.2028443841i\)
\(L(\frac12)\) \(\approx\) \(-0.3012752972 - 0.2028443841i\)
\(L(1)\) \(\approx\) \(0.4523041207 - 0.5821237424i\)
\(L(1)\) \(\approx\) \(0.4523041207 - 0.5821237424i\)

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 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (0.207 - 0.978i)T \)
11 \( 1 + (0.838 + 0.544i)T \)
13 \( 1 + (-0.156 - 0.987i)T \)
17 \( 1 + (-0.544 + 0.838i)T \)
19 \( 1 + (-0.777 - 0.629i)T \)
23 \( 1 + (-0.913 + 0.406i)T \)
29 \( 1 + (-0.453 + 0.891i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (0.358 - 0.933i)T \)
53 \( 1 + (-0.998 + 0.0523i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (-0.994 + 0.104i)T \)
67 \( 1 + (-0.0523 - 0.998i)T \)
71 \( 1 + (0.891 - 0.453i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.965 - 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (0.629 - 0.777i)T \)
97 \( 1 + (-0.891 - 0.453i)T \)
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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

−26.226702593240164729040172463324, −25.31046879529818421631390705892, −24.50040712785543810167014423285, −23.25352351834302829124309744899, −22.36429263055418915454937982190, −21.871531961076067877787332329903, −20.64872801205085666178775391845, −19.32252540195632169758243291052, −18.85742410352160365362924734780, −17.56101862570969135669962495380, −16.801677301296160387972422187075, −15.90293968544243664461438728770, −15.06937954577664140393197198054, −14.09349891120475689677921682693, −13.87370846321253006338402329480, −11.700939569721392870121253888399, −10.718369428418423723252635311819, −9.81706100113739398882424010565, −9.05218782617513572439319944460, −8.032182306049041801115597637051, −6.7034009771127584868026533099, −6.02267938578222985472714583801, −4.615757037922249379651808174710, −3.68807278173032343372759484659, −2.1041274766627790160324535823, 0.12204435067015343766404625632, 1.35844194310082737848325285604, 2.16289045217826517816501539954, 3.60360804200958800813326725516, 4.82935808465586002343591995192, 6.26124182575925796018269768291, 7.63112342656286787740627081377, 8.5246385981444983130518898723, 9.20981957456044128050561717916, 10.39506128311421178838585365001, 11.69015125555097843035262580569, 12.48113843273603913025834551371, 13.04474050753764932943270562225, 13.93849955451994385981196393427, 15.24297668450979134045045647152, 16.8747905146254658485029464204, 17.46261505080372182593344784932, 18.11646290292192326284225317367, 19.54342117101297614325647529154, 19.78700025705777265497667912815, 20.61656276221142934358934965906, 21.71234281561146005349368477554, 22.687086998433456300484161208079, 23.73679610930245452181250552530, 24.673309518278861945839359231692

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