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Properties

Label 1-127-127.15-r0-0-0
Degree $1$
Conductor $127$
Sign $-0.573 - 0.819i$
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 + (−0.318 − 0.947i)3-s + (−0.900 − 0.433i)4-s + (0.0747 − 0.997i)5-s + (0.995 − 0.0995i)6-s + (−0.411 + 0.911i)7-s + (0.623 − 0.781i)8-s + (−0.797 + 0.603i)9-s + (0.955 + 0.294i)10-s + (−0.998 + 0.0498i)11-s + (−0.124 + 0.992i)12-s + (−0.583 − 0.811i)13-s + (−0.797 − 0.603i)14-s + (−0.969 + 0.246i)15-s + (0.623 + 0.781i)16-s + (−0.969 − 0.246i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.318 − 0.947i)3-s + (−0.900 − 0.433i)4-s + (0.0747 − 0.997i)5-s + (0.995 − 0.0995i)6-s + (−0.411 + 0.911i)7-s + (0.623 − 0.781i)8-s + (−0.797 + 0.603i)9-s + (0.955 + 0.294i)10-s + (−0.998 + 0.0498i)11-s + (−0.124 + 0.992i)12-s + (−0.583 − 0.811i)13-s + (−0.797 − 0.603i)14-s + (−0.969 + 0.246i)15-s + (0.623 + 0.781i)16-s + (−0.969 − 0.246i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $-0.573 - 0.819i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ -0.573 - 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1417012080 - 0.2719815479i\)
\(L(\frac12)\) \(\approx\) \(0.1417012080 - 0.2719815479i\)
\(L(1)\) \(\approx\) \(0.5377986882 - 0.06597034110i\)
\(L(1)\) \(\approx\) \(0.5377986882 - 0.06597034110i\)

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 \( 1 + (-0.318 - 0.947i)T \)
5 \( 1 + (0.0747 - 0.997i)T \)
7 \( 1 + (-0.411 + 0.911i)T \)
11 \( 1 + (-0.998 + 0.0498i)T \)
13 \( 1 + (-0.583 - 0.811i)T \)
17 \( 1 + (-0.969 - 0.246i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.998 - 0.0498i)T \)
29 \( 1 + (0.878 - 0.478i)T \)
31 \( 1 + (0.698 + 0.715i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (0.698 - 0.715i)T \)
43 \( 1 + (-0.853 - 0.521i)T \)
47 \( 1 + (0.826 - 0.563i)T \)
53 \( 1 + (-0.124 - 0.992i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.988 + 0.149i)T \)
67 \( 1 + (0.980 + 0.198i)T \)
71 \( 1 + (0.456 + 0.889i)T \)
73 \( 1 + (-0.733 + 0.680i)T \)
79 \( 1 + (0.542 - 0.840i)T \)
83 \( 1 + (0.921 + 0.388i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
97 \( 1 + (-0.0249 - 0.999i)T \)
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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

−29.197334079773734423640675907320, −28.30599643164590734231545895172, −27.01679876141595863395834131789, −26.46803958869756356585934895665, −25.93772119810788748824102880981, −23.6008893846536059970524968196, −22.84975338472383134708918588050, −21.94719327154090694352485602768, −21.19916185752602713465396233958, −20.11432552214214535005308493211, −19.15122415223396249597328893123, −17.96824866627654144681145602080, −17.05649487806693779238942639585, −15.90163514881884932827694545079, −14.4931874517796403279092071519, −13.59833955461792516574969675523, −12.116276249514524527343939011821, −10.8793816486468967545486941240, −10.37347256028393046318910893005, −9.51009824285997143971612680416, −7.93097166046640620951861841296, −6.34943008426085958294556945899, −4.61294918964809217099009553677, −3.65910595914051108988742007644, −2.41233877406711523058891513245, 0.29688772946615155202510317474, 2.332598680301008333436824240696, 4.87965562483903285297938789198, 5.640859986116137582725667183033, 6.80303890771421971665779113416, 8.12343937352484869967876554341, 8.785793034519969343206187144966, 10.2298652264160368474624884629, 12.09335831647743799852290464879, 12.94585286426648082072374139064, 13.69507199147356156431815896108, 15.39336429962111747431447525465, 16.04365109702370897723233631751, 17.42554949413511580612512296780, 17.89450879691404326186131221501, 19.0951239978360021872261454216, 19.99395663649463882440847891044, 21.72772305740129831558493635410, 22.76070543360702018700672107586, 23.817495330701613423325386039586, 24.52850432951955815871480875108, 25.19513492082137843981159296662, 26.15212657860707130389542234598, 27.64808462304051258825259211626, 28.494292194864778264840273366076

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