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Properties

Label 1-247-247.200-r0-0-0
Degree $1$
Conductor $247$
Sign $0.311 - 0.950i$
Analytic cond. $1.14706$
Root an. cond. $1.14706$
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  + (−0.342 − 0.939i)2-s + (−0.173 + 0.984i)3-s + (−0.766 + 0.642i)4-s + (0.642 − 0.766i)5-s + (0.984 − 0.173i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (−0.939 − 0.342i)9-s + (−0.939 − 0.342i)10-s + (−0.866 − 0.5i)11-s + (−0.5 − 0.866i)12-s + (−0.766 − 0.642i)14-s + (0.642 + 0.766i)15-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + i·18-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.173 + 0.984i)3-s + (−0.766 + 0.642i)4-s + (0.642 − 0.766i)5-s + (0.984 − 0.173i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (−0.939 − 0.342i)9-s + (−0.939 − 0.342i)10-s + (−0.866 − 0.5i)11-s + (−0.5 − 0.866i)12-s + (−0.766 − 0.642i)14-s + (0.642 + 0.766i)15-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(247\)    =    \(13 \cdot 19\)
Sign: $0.311 - 0.950i$
Analytic conductor: \(1.14706\)
Root analytic conductor: \(1.14706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{247} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 247,\ (0:\ ),\ 0.311 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8078098853 - 0.5850531127i\)
\(L(\frac12)\) \(\approx\) \(0.8078098853 - 0.5850531127i\)
\(L(1)\) \(\approx\) \(0.8547711112 - 0.3285561256i\)
\(L(1)\) \(\approx\) \(0.8547711112 - 0.3285561256i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.342 - 0.939i)T \)
3 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (0.642 - 0.766i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.984 + 0.173i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (0.342 - 0.939i)T \)
53 \( 1 + (-0.766 + 0.642i)T \)
59 \( 1 + (0.342 + 0.939i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (-0.642 + 0.766i)T \)
73 \( 1 + (0.984 + 0.173i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (0.984 - 0.173i)T \)
97 \( 1 + (-0.342 - 0.939i)T \)
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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

−25.94028011800405102996227583814, −25.34886194518417427784227240434, −24.549941195018506412241288957486, −23.65769948362496513673349457382, −22.94265699899930391852028582278, −21.93450960819976199283144949784, −20.75806279950785693681103429105, −19.21586705356205200499349668232, −18.55478232490687622823546062666, −17.798387269404529992970680043372, −17.36822450827438603386241644913, −15.9784132466829068022579490269, −14.75358859490041037288372487754, −14.22807706529311839928059537915, −13.24224586237135883491486832165, −12.10272337822520569628140256989, −10.75903323457611447540128201950, −9.84805882142134463371349406142, −8.32795114077196500326277400974, −7.771237987516074851059870852213, −6.62853460483460279990751402184, −5.82123990964724292257158196207, −4.86834970744306521175307559310, −2.63656695289723773813033184533, −1.45907739850096971156823947370, 0.91951007114168430935076780845, 2.44909398383267355467763434365, 3.81509405445312795061724444323, 4.86421785836864796989598709309, 5.59138923290752271222044069899, 7.8706115701659972486615959842, 8.64603395687133999209260729031, 9.77772225743709428698877854506, 10.391029469687483581884784963864, 11.372137995631910240161141323578, 12.32184355376742875100123916141, 13.63997548071059786404387353834, 14.239544396125643489272921779193, 15.866254289622757211521858556880, 16.75635571377515846798334963178, 17.49756762170789241222374225488, 18.32005599555635258234553511236, 19.7882692967892271072289055110, 20.57509574790383501502169463249, 21.26634342374873081761770617432, 21.6289977758068774460402274297, 23.00198719953592877840684074420, 23.8296769972884361474110543682, 25.25255338910098471418515792317, 26.25681097985123806826918071616

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