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Properties

Label 1-127-127.21-r0-0-0
Degree $1$
Conductor $127$
Sign $0.867 + 0.497i$
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  + (−0.222 + 0.974i)2-s + (0.878 − 0.478i)3-s + (−0.900 − 0.433i)4-s + (0.826 + 0.563i)5-s + (0.270 + 0.962i)6-s + (0.698 − 0.715i)7-s + (0.623 − 0.781i)8-s + (0.542 − 0.840i)9-s + (−0.733 + 0.680i)10-s + (−0.797 − 0.603i)11-s + (−0.998 + 0.0498i)12-s + (−0.969 − 0.246i)13-s + (0.542 + 0.840i)14-s + (0.995 + 0.0995i)15-s + (0.623 + 0.781i)16-s + (0.995 − 0.0995i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (0.878 − 0.478i)3-s + (−0.900 − 0.433i)4-s + (0.826 + 0.563i)5-s + (0.270 + 0.962i)6-s + (0.698 − 0.715i)7-s + (0.623 − 0.781i)8-s + (0.542 − 0.840i)9-s + (−0.733 + 0.680i)10-s + (−0.797 − 0.603i)11-s + (−0.998 + 0.0498i)12-s + (−0.969 − 0.246i)13-s + (0.542 + 0.840i)14-s + (0.995 + 0.0995i)15-s + (0.623 + 0.781i)16-s + (0.995 − 0.0995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.867 + 0.497i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.867 + 0.497i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.273487436 + 0.3393597698i\)
\(L(\frac12)\) \(\approx\) \(1.273487436 + 0.3393597698i\)
\(L(1)\) \(\approx\) \(1.221144293 + 0.3037717338i\)
\(L(1)\) \(\approx\) \(1.221144293 + 0.3037717338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad127 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 + (0.878 - 0.478i)T \)
5 \( 1 + (0.826 + 0.563i)T \)
7 \( 1 + (0.698 - 0.715i)T \)
11 \( 1 + (-0.797 - 0.603i)T \)
13 \( 1 + (-0.969 - 0.246i)T \)
17 \( 1 + (0.995 - 0.0995i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.797 + 0.603i)T \)
29 \( 1 + (0.980 + 0.198i)T \)
31 \( 1 + (-0.583 + 0.811i)T \)
37 \( 1 + (0.173 + 0.984i)T \)
41 \( 1 + (-0.583 - 0.811i)T \)
43 \( 1 + (-0.661 + 0.749i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (-0.998 - 0.0498i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (0.365 - 0.930i)T \)
67 \( 1 + (-0.853 - 0.521i)T \)
71 \( 1 + (-0.124 - 0.992i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + (0.921 + 0.388i)T \)
83 \( 1 + (0.456 + 0.889i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
97 \( 1 + (-0.318 + 0.947i)T \)
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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

−28.474093028983686420019192857350, −28.00409502009272182183591615601, −26.85226858609263005027538551206, −25.86979880727176364081536807670, −25.03243071464010997851231303068, −23.785073793711205395177554726776, −22.01245324345217741806379697455, −21.45971560682733006502254587643, −20.712534809338177057326600862821, −19.86261302798795209768036127088, −18.66034347922745226341791055625, −17.73434117337102935967748725554, −16.58393752126206983360940201630, −15.01142170243712769669311764122, −14.09474908175692690756680565533, −12.987186162866365675901684295048, −12.065175064272017026119927487253, −10.44500782459081661596662569785, −9.687678375753140167834097859159, −8.73270174868289747543094161128, −7.78759632594578556131849545944, −5.24743974928064312670775163509, −4.47408516035167748817188159393, −2.60969148510984215746930178693, −1.94422744975849705799157860782, 1.57543675627337276505719319551, 3.30277024443749224802665083622, 5.051752686649118025490000277590, 6.36583062038655321817059603647, 7.57001358066892367712527654727, 8.18037930496759232303053800329, 9.701850189377330147595179157751, 10.4590923707040126730368462732, 12.61037164882400827068073227188, 13.92225842048038254330592845442, 14.18047505393705453812698427358, 15.23245609020858529548491242975, 16.70136813122969261220020924699, 17.744282305070851521689652515949, 18.47818663111809448925192050856, 19.4721947173537047164387707148, 20.858708455992595710122266222439, 21.82944127076821586009426768115, 23.34932984175992062013607467661, 24.03397505218228263147293667932, 25.12259224012084826212959173282, 25.737306015441118723653793605893, 26.750175522942109882491418444211, 27.31144222301507870088435841100, 29.16754561431573403753650441326

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