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Properties

Label 1-287-287.220-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.341 + 0.939i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.341 + 0.939i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (220, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.341 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7004876039 + 0.9994434789i\)
\(L(\frac12)\) \(\approx\) \(0.7004876039 + 0.9994434789i\)
\(L(1)\) \(\approx\) \(0.8813522368 + 0.5848748683i\)
\(L(1)\) \(\approx\) \(0.8813522368 + 0.5848748683i\)

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

−25.58128879630375655069404825038, −24.51632832944779127628099444576, −23.81736072287037416270488566628, −22.12302610326437481505435722697, −21.227884173773966353418887055579, −20.54825778211533962183836032327, −19.9281742836033851865967433225, −19.02171230518903324979502531360, −18.022535555134124261089718789202, −17.36520386923821716436353312810, −16.0393990749668554071887225147, −15.37424106162225718227465577077, −13.59283870141960736254079176751, −13.19416231776058180533513918753, −12.36191435532077793649111476274, −10.996674818641531480676513261486, −9.91956263949718219951927134972, −9.13230098201810308134823646448, −8.19218623942082824111570238177, −7.66452696347728835239700479298, −5.97095295095865539486458660861, −4.39687189483616645046201446777, −3.18555877664112168794004166032, −2.14933559568716130861361464216, −0.97258471607896381309107090722, 1.83293453395565884377918142765, 2.75853794584651956816557161344, 4.3349184841229197376632236570, 5.708821867406058180248049029998, 6.93383436619707414722623232364, 7.64034265703837824226048297683, 8.70892915526950601536462967497, 9.733804131503350093835938820046, 10.29673915856851982115803736845, 11.40999627989648574910262902121, 13.26849595820023023805873385475, 14.138836665137360629345536938654, 14.701471988019077916660935556673, 15.78172664075487324446000230322, 16.36206602540254088074805136621, 17.90247392372109588526949125323, 18.48162686501556617339544194354, 19.13657117254644443403769144200, 20.35307583597559924925231830377, 21.00682651490414715004126773819, 22.28028614391859031019177134592, 23.22941832042634818141898363843, 24.35554163324490568318914506762, 25.16210248801772720430117235770, 25.98448029313238319084501958

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