Properties

Label 1-287-287.171-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.658 + 0.752i$
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  + (−0.406 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.658 + 0.752i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1097137808 - 0.2419361870i\)
\(L(\frac12)\) \(\approx\) \(-0.1097137808 - 0.2419361870i\)
\(L(1)\) \(\approx\) \(0.4611128433 - 0.3715229467i\)
\(L(1)\) \(\approx\) \(0.4611128433 - 0.3715229467i\)

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

−25.98965574265559300283224686062, −25.44858072822850320659259354775, −24.32579421551761314529345077670, −23.600430395436024398053965495506, −22.65553744798833801904572493197, −21.57545015727630314761946719854, −20.56423476993343238702722967735, −19.87016520538478350186943896066, −18.7407452895997500341021035699, −17.62109122788951347046293215570, −16.59114284474679429236154743655, −16.14261626056454128433711119574, −15.32194810480399990840072944718, −14.34015440795954495748858499958, −13.43356747240216562605229474105, −12.158429506772263110783722007499, −10.72901436973999336897529217838, −9.798073661552479045056690894862, −8.94065907542182351716809507057, −8.26412403429606400176473550596, −7.09537368963249176228636448833, −5.619599829044113406929511902399, −4.76509641071875095645874612315, −4.04629139657935046454689613164, −2.016564071549955269422715145646, 0.18305217346598808751040019197, 2.00622056147039777765786945678, 2.81264349464204494211376253067, 3.8052447993067846299282054195, 5.64068105374714593475412839435, 6.92973600332364691187763239542, 7.95671366191927154650991946775, 8.56130328811808428742573447853, 10.10794594543023878916201334293, 10.87731889658584103735816307456, 11.751775999900016312843685463488, 12.87803403975400043480198705456, 13.46026465669837757643732645020, 14.55299388824674942142810508194, 15.659165497571732171382759833681, 17.30375282243148644819871711968, 17.91620972177448783157676502913, 18.66980679334659882069193795605, 19.47677968936799149594054263121, 20.046504782660543163646554494678, 21.32081087018997095418472423108, 22.144565826638977270348033651725, 23.135553395263000825712626816472, 23.85873702753199977635173498772, 25.1953021244193283057373162405

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