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Properties

Label 1-287-287.219-r1-0-0
Degree $1$
Conductor $287$
Sign $0.920 + 0.389i$
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  + (0.866 − 0.5i)2-s + (−0.258 − 0.965i)3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.707 − 0.707i)6-s − i·8-s + (−0.866 + 0.5i)9-s + (−0.5 + 0.866i)10-s + (0.258 + 0.965i)11-s + (−0.965 − 0.258i)12-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)15-s + (−0.5 − 0.866i)16-s + (0.965 − 0.258i)17-s + (−0.5 + 0.866i)18-s + (−0.258 + 0.965i)19-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.258 − 0.965i)3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.707 − 0.707i)6-s − i·8-s + (−0.866 + 0.5i)9-s + (−0.5 + 0.866i)10-s + (0.258 + 0.965i)11-s + (−0.965 − 0.258i)12-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)15-s + (−0.5 − 0.866i)16-s + (0.965 − 0.258i)17-s + (−0.5 + 0.866i)18-s + (−0.258 + 0.965i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.920 + 0.389i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.920 + 0.389i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.539822391 + 0.3123758990i\)
\(L(\frac12)\) \(\approx\) \(1.539822391 + 0.3123758990i\)
\(L(1)\) \(\approx\) \(1.163307684 - 0.4022875300i\)
\(L(1)\) \(\approx\) \(1.163307684 - 0.4022875300i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.258 + 0.965i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (0.965 - 0.258i)T \)
19 \( 1 + (-0.258 + 0.965i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.258 + 0.965i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (0.965 - 0.258i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (0.965 + 0.258i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
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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

−25.04192134414855723110224741572, −24.196220478053839369325239651214, −23.36658156119858307819141211722, −22.56665574718474265156802715381, −21.79466481017058355699967257015, −20.91521933812108763270477353682, −20.14291558366820262421359741837, −19.12720799449188348861461905214, −17.3188844980939751325394821401, −16.76497356811441816621676581591, −15.92529891167802960996776465285, −15.171210548779664302543024801823, −14.447675499379417999054074873868, −13.166141346607627568569172901216, −12.11085284574402843583016565877, −11.423910994139312032511466240504, −10.37199558857045173375413491419, −8.823050959301984384796250194767, −8.120168175518785373402984456324, −6.800178470920038184539399427982, −5.50487897105981969750793409730, −4.830643530548653315967802422806, −3.75338712982802854959510904540, −2.95763604689861318364500982032, −0.38089346294826506293919009100, 1.30568831327613641713123082453, 2.44236143574365217818340976559, 3.64188236162201160995710477342, 4.81590554609318663259773126333, 6.02204778464703144140739963952, 7.12616860188866640779049876847, 7.65432608389280255564898355270, 9.47917391635327186830185790343, 10.68627028434772266929863768609, 11.74112603463815219011958632587, 12.15363665313817641142849370022, 13.05414421089251848122405089264, 14.40423751842012132533013501648, 14.6933808908861782950860505240, 16.045515592216438184366731354054, 17.14766619782656457812721040604, 18.494153001664783386430140753918, 19.109294104373773672361536403881, 19.80062675607569737562946413945, 20.78100000038202360135425176453, 22.012616013662468111991837053699, 22.98131428450429822777243161905, 23.23248365589934721919131060252, 24.22645087194210172473023838230, 25.04333228954563735430500186611

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