Properties

Label 1-287-287.88-r1-0-0
Degree $1$
Conductor $287$
Sign $0.792 - 0.609i$
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  + (0.994 − 0.104i)2-s + (0.965 + 0.258i)3-s + (0.978 − 0.207i)4-s + (0.743 − 0.669i)5-s + (0.987 + 0.156i)6-s + (0.951 − 0.309i)8-s + (0.866 + 0.5i)9-s + (0.669 − 0.743i)10-s + (−0.0523 − 0.998i)11-s + (0.998 + 0.0523i)12-s + (0.156 − 0.987i)13-s + (0.891 − 0.453i)15-s + (0.913 − 0.406i)16-s + (−0.998 + 0.0523i)17-s + (0.913 + 0.406i)18-s + (−0.933 − 0.358i)19-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)2-s + (0.965 + 0.258i)3-s + (0.978 − 0.207i)4-s + (0.743 − 0.669i)5-s + (0.987 + 0.156i)6-s + (0.951 − 0.309i)8-s + (0.866 + 0.5i)9-s + (0.669 − 0.743i)10-s + (−0.0523 − 0.998i)11-s + (0.998 + 0.0523i)12-s + (0.156 − 0.987i)13-s + (0.891 − 0.453i)15-s + (0.913 − 0.406i)16-s + (−0.998 + 0.0523i)17-s + (0.913 + 0.406i)18-s + (−0.933 − 0.358i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.792 - 0.609i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.792 - 0.609i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.834541935 - 1.985215509i\)
\(L(\frac12)\) \(\approx\) \(5.834541935 - 1.985215509i\)
\(L(1)\) \(\approx\) \(2.997949960 - 0.5482150881i\)
\(L(1)\) \(\approx\) \(2.997949960 - 0.5482150881i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.994 + 0.104i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (-0.743 + 0.669i)T \)
11 \( 1 + (0.0523 + 0.998i)T \)
13 \( 1 + (-0.156 + 0.987i)T \)
17 \( 1 + (0.998 - 0.0523i)T \)
19 \( 1 + (0.933 + 0.358i)T \)
23 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (-0.453 - 0.891i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (-0.669 - 0.743i)T \)
43 \( 1 + (0.587 + 0.809i)T \)
47 \( 1 + (0.629 - 0.777i)T \)
53 \( 1 + (0.544 - 0.838i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
61 \( 1 + (-0.406 - 0.913i)T \)
67 \( 1 + (-0.838 - 0.544i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.258 - 0.965i)T \)
83 \( 1 - T \)
89 \( 1 + (0.358 - 0.933i)T \)
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

−25.25633905574900671278012803427, −24.645939022872544098242747085058, −23.589332403797391884849920385795, −22.692204561006308316308516490108, −21.67878646583024080774559354141, −21.02347475801222564315797021496, −20.18831142060253614627027839791, −19.200943697215686639752301290904, −18.222899689844635209313704274061, −17.103694681965363150421908455273, −15.84018217032165350764322255210, −14.726515454888782327110213605242, −14.50034985251100426956117367531, −13.32786179211371687329438048637, −12.8179964798626099942710047759, −11.50478086868675487121646229685, −10.34987281959601584963349374090, −9.34577269014594829299127821719, −8.016517875871954553542078208133, −6.79994275859903726010295058363, −6.40519231310275520791621573546, −4.68510763798104639369235409477, −3.78595651777408432162824859911, −2.30695985791047774983541878128, −2.00904658476596928426094270673, 1.29058926515042844058753328390, 2.49304148264805172579381717493, 3.4599461821656915876101331734, 4.654068816276338056229398160363, 5.581736774963415746107977561351, 6.74069470458458388049075003123, 8.146381106150492265988650711621, 8.9907660739495840852174396304, 10.24246600017029000900788925684, 11.07258004605455711307460119616, 12.637111005377221212344153409616, 13.2786503427090747000619885487, 13.87298563275358029731388329286, 14.94229143443765853979340302288, 15.76694168559296764214564845174, 16.61666518188055586202097932461, 17.88649557807475513544309076617, 19.321115479041018365868513972921, 20.02529034518121295619854945592, 20.77886554237575066444996215631, 21.67707557190953839862360249887, 22.04670722450528414390632028449, 23.659426351872702367171722198084, 24.26987824080375011175447518470, 25.32925567992916335996390314851

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