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Properties

Label 1-287-287.271-r1-0-0
Degree $1$
Conductor $287$
Sign $0.100 + 0.994i$
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  + (−0.104 + 0.994i)2-s + (−0.5 + 0.866i)3-s + (−0.978 − 0.207i)4-s + (−0.669 + 0.743i)5-s + (−0.809 − 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.5 − 0.866i)9-s + (−0.669 − 0.743i)10-s + (−0.669 − 0.743i)11-s + (0.669 − 0.743i)12-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)15-s + (0.913 + 0.406i)16-s + (0.669 + 0.743i)17-s + (0.913 − 0.406i)18-s + (0.913 + 0.406i)19-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)2-s + (−0.5 + 0.866i)3-s + (−0.978 − 0.207i)4-s + (−0.669 + 0.743i)5-s + (−0.809 − 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.5 − 0.866i)9-s + (−0.669 − 0.743i)10-s + (−0.669 − 0.743i)11-s + (0.669 − 0.743i)12-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)15-s + (0.913 + 0.406i)16-s + (0.669 + 0.743i)17-s + (0.913 − 0.406i)18-s + (0.913 + 0.406i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.100 + 0.994i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.100 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5340196220 + 0.4825731064i\)
\(L(\frac12)\) \(\approx\) \(0.5340196220 + 0.4825731064i\)
\(L(1)\) \(\approx\) \(0.4650637317 + 0.4127307081i\)
\(L(1)\) \(\approx\) \(0.4650637317 + 0.4127307081i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.104 + 0.994i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.669 + 0.743i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 + (-0.104 + 0.994i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (0.978 + 0.207i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−24.89417960367448687163910704137, −23.9476099785308383381476937243, −23.254473286881642492396040932996, −22.48766417209906406229911702334, −21.41791429879664188242865387194, −20.15775992915853269178202504622, −19.90019847337647822049722011148, −18.60909783919491238135197101454, −18.183794011882367564676098498780, −16.96292965681585892888001804879, −16.290019901364702130301985055647, −14.662960745390852730753764244252, −13.55104665012123843925205155544, −12.614892014084521182707219940626, −12.09109609043617937520927712650, −11.32825509973959240369515785780, −10.1163374791552472149429697751, −9.01271299015553288470740382143, −7.88327526350472590063798314711, −7.12607777259973481165234100614, −5.19558640319532737517796862710, −4.73235036815578131671935758038, −3.06364916435883088137471613702, −1.82995863294398717856732692521, −0.649463894365034327115112606655, 0.43655638303192094334047486256, 3.18613330295591635193606719882, 4.048218740802978927462505700985, 5.39154528569033175720304036888, 5.98639508268447214322385419953, 7.43947195125607784768076527258, 8.08622072811407761769651021470, 9.52720146538391859037381848419, 10.28667632699789406792356247772, 11.272872001980333553447861431197, 12.43817200978346007978557231456, 13.82848428999517063371361395823, 14.85803580252709406809120083941, 15.388904035653427600115383242810, 16.242413033615601151173309142084, 17.057310843655362339450957085525, 18.05552529519411345657866056147, 18.887469522770614098468115731759, 19.959601283199765667693352725231, 21.41187503343763645229795699324, 22.11114156368552023711648642982, 22.9395321794466037585854553206, 23.577954618068455556081113928287, 24.49923764093856208745750698826, 25.81675317949098685792020955581

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