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Properties

Label 1-287-287.194-r0-0-0
Degree $1$
Conductor $287$
Sign $0.698 - 0.715i$
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  + (−0.743 − 0.669i)2-s + (−0.965 + 0.258i)3-s + (0.104 + 0.994i)4-s + (0.406 − 0.913i)5-s + (0.891 + 0.453i)6-s + (0.587 − 0.809i)8-s + (0.866 − 0.5i)9-s + (−0.913 + 0.406i)10-s + (0.933 + 0.358i)11-s + (−0.358 − 0.933i)12-s + (−0.453 + 0.891i)13-s + (−0.156 + 0.987i)15-s + (−0.978 + 0.207i)16-s + (0.358 − 0.933i)17-s + (−0.978 − 0.207i)18-s + (−0.544 + 0.838i)19-s + ⋯
L(s)  = 1  + (−0.743 − 0.669i)2-s + (−0.965 + 0.258i)3-s + (0.104 + 0.994i)4-s + (0.406 − 0.913i)5-s + (0.891 + 0.453i)6-s + (0.587 − 0.809i)8-s + (0.866 − 0.5i)9-s + (−0.913 + 0.406i)10-s + (0.933 + 0.358i)11-s + (−0.358 − 0.933i)12-s + (−0.453 + 0.891i)13-s + (−0.156 + 0.987i)15-s + (−0.978 + 0.207i)16-s + (0.358 − 0.933i)17-s + (−0.978 − 0.207i)18-s + (−0.544 + 0.838i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.698 - 0.715i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.698 - 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6352667233 - 0.2677605270i\)
\(L(\frac12)\) \(\approx\) \(0.6352667233 - 0.2677605270i\)
\(L(1)\) \(\approx\) \(0.6247339838 - 0.1829990338i\)
\(L(1)\) \(\approx\) \(0.6247339838 - 0.1829990338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.743 - 0.669i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (0.406 - 0.913i)T \)
11 \( 1 + (0.933 + 0.358i)T \)
13 \( 1 + (-0.453 + 0.891i)T \)
17 \( 1 + (0.358 - 0.933i)T \)
19 \( 1 + (-0.544 + 0.838i)T \)
23 \( 1 + (-0.669 + 0.743i)T \)
29 \( 1 + (0.987 + 0.156i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (0.951 - 0.309i)T \)
47 \( 1 + (-0.0523 - 0.998i)T \)
53 \( 1 + (0.629 - 0.777i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (-0.207 - 0.978i)T \)
67 \( 1 + (0.777 + 0.629i)T \)
71 \( 1 + (0.156 + 0.987i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.258 - 0.965i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.838 - 0.544i)T \)
97 \( 1 + (0.156 - 0.987i)T \)
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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

−25.5874515848672068338451893377, −24.79299095463666951923351769282, −23.97033439998207574775244522281, −22.98087356053625958918423020944, −22.30380501505429545337267464922, −21.41077635918230026016941327381, −19.661377516094492407694500611073, −19.12448851283156273090359626280, −18.00444014859598540692988609519, −17.51259205369313121106961253779, −16.773139787591514129660721812826, −15.650466294731553218482349505806, −14.74830866145148784928163391955, −13.830246212738966083380868812057, −12.49675243991713868814874959459, −11.2577803444281657661898806699, −10.50920549797929808251408814436, −9.78943912084811621408687116410, −8.36140296366168366882069121836, −7.26305169742838026608852457671, −6.33580537669473955272562545550, −5.855711366273656441149899726330, −4.43889342984481630996833764105, −2.46216272606600033776044940149, −0.99558747741666631332953594525, 0.94162027543308038050791885271, 2.02477904641913729190270403555, 3.9479028726710669797669457158, 4.73885302410278646435532416345, 6.14305438903019710964054751385, 7.22987861410767988055684558182, 8.597857953931220575718841924518, 9.672316704582227466808203320565, 10.03594320034374507691070996286, 11.66982087285721443048475813087, 11.90238727303534800150121973123, 12.9032618842802510746655102532, 14.13973757930361327923401238533, 15.8184533302021054283427823088, 16.62471073741385565423124235453, 17.15991261563604200205065606292, 17.95163218620155238345227657705, 19.01262187561651435398199342214, 20.01434602121197595480428551117, 20.97891099609708058641031013011, 21.596138720158426787421896454874, 22.46467005356223692697496256167, 23.54250459209468069187173238362, 24.68737587932944441823183918916, 25.392357436061132863621636518950

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