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Properties

Label 1-287-287.240-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: $\chi_{287} (240, \cdot )$
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

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

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