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Properties

Label 1-127-127.11-r0-0-0
Degree $1$
Conductor $127$
Sign $-0.363 - 0.931i$
Analytic cond. $0.589785$
Root an. cond. $0.589785$
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  + (0.623 − 0.781i)2-s + (−0.969 − 0.246i)3-s + (−0.222 − 0.974i)4-s + (0.955 − 0.294i)5-s + (−0.797 + 0.603i)6-s + (0.921 + 0.388i)7-s + (−0.900 − 0.433i)8-s + (0.878 + 0.478i)9-s + (0.365 − 0.930i)10-s + (−0.318 − 0.947i)11-s + (−0.0249 + 0.999i)12-s + (−0.124 − 0.992i)13-s + (0.878 − 0.478i)14-s + (−0.998 + 0.0498i)15-s + (−0.900 + 0.433i)16-s + (−0.998 − 0.0498i)17-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.969 − 0.246i)3-s + (−0.222 − 0.974i)4-s + (0.955 − 0.294i)5-s + (−0.797 + 0.603i)6-s + (0.921 + 0.388i)7-s + (−0.900 − 0.433i)8-s + (0.878 + 0.478i)9-s + (0.365 − 0.930i)10-s + (−0.318 − 0.947i)11-s + (−0.0249 + 0.999i)12-s + (−0.124 − 0.992i)13-s + (0.878 − 0.478i)14-s + (−0.998 + 0.0498i)15-s + (−0.900 + 0.433i)16-s + (−0.998 − 0.0498i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $-0.363 - 0.931i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ -0.363 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7040157753 - 1.030374095i\)
\(L(\frac12)\) \(\approx\) \(0.7040157753 - 1.030374095i\)
\(L(1)\) \(\approx\) \(0.9709614035 - 0.7380646555i\)
\(L(1)\) \(\approx\) \(0.9709614035 - 0.7380646555i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad127 \( 1 \)
good2 \( 1 + (0.623 - 0.781i)T \)
3 \( 1 + (-0.969 - 0.246i)T \)
5 \( 1 + (0.955 - 0.294i)T \)
7 \( 1 + (0.921 + 0.388i)T \)
11 \( 1 + (-0.318 - 0.947i)T \)
13 \( 1 + (-0.124 - 0.992i)T \)
17 \( 1 + (-0.998 - 0.0498i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.318 + 0.947i)T \)
29 \( 1 + (0.995 - 0.0995i)T \)
31 \( 1 + (0.456 - 0.889i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (0.456 + 0.889i)T \)
43 \( 1 + (-0.411 + 0.911i)T \)
47 \( 1 + (-0.733 + 0.680i)T \)
53 \( 1 + (-0.0249 - 0.999i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (0.826 + 0.563i)T \)
67 \( 1 + (0.270 + 0.962i)T \)
71 \( 1 + (-0.661 - 0.749i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 + (0.980 - 0.198i)T \)
83 \( 1 + (-0.853 + 0.521i)T \)
89 \( 1 + (0.365 + 0.930i)T \)
97 \( 1 + (-0.583 + 0.811i)T \)
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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

−29.15357874717761937223581922304, −28.22533596223961383851158801055, −26.83950126780319017650281931981, −26.2016871369889877028096249422, −24.91358127270296594262036827061, −23.96696554559516759138587088261, −23.23252290202500697933697462019, −22.06321233497659242323327173437, −21.49512125592202618912792233295, −20.53017067995029162255077141099, −18.278983123940119096438375400265, −17.56977921728768456213846197150, −16.98925373415895644072248669768, −15.70752881829016169854284913474, −14.67020237040211796461461713625, −13.668010829038060456849351196677, −12.53716782101632634670301112725, −11.355342684249205615070789745973, −10.22413548513289077976753711404, −8.79428631213441868226539652658, −7.04386229109019395517059194292, −6.444841087096345086177755451189, −4.95245496151606033852152322709, −4.44133057591582092528714751827, −2.15432954154219265998353496241, 1.20568952259018479548296619809, 2.47355828615453413697510965677, 4.50597269719677471782283683305, 5.551382567991336136636177673381, 6.15546547822109654583678300310, 8.23676322955405734325816023989, 9.81776981968166932931855678973, 10.82134678157442870234737607163, 11.64523268335421672767628059668, 12.855953989263232729333389362455, 13.54361978845075557766388751901, 14.83336537296365830454932441647, 16.16497676380254135149957562682, 17.69499004181676374848089209192, 18.06657848066273110224011910337, 19.37508451450057963392429406433, 20.83431431184039902546141766647, 21.51041834153554351229977425995, 22.24339839465552119977406609776, 23.40075294505499405717757198139, 24.39008882569280676057354717941, 24.94963542942905893613501602165, 27.012781269241785818300776688009, 27.8611461214328512554443884606, 28.72786028990405477716451255820

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