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Properties

Label 1-127-127.69-r0-0-0
Degree $1$
Conductor $127$
Sign $0.857 - 0.515i$
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 + (0.980 − 0.198i)3-s + (−0.900 + 0.433i)4-s + (0.0747 + 0.997i)5-s + (−0.411 − 0.911i)6-s + (−0.583 + 0.811i)7-s + (0.623 + 0.781i)8-s + (0.921 − 0.388i)9-s + (0.955 − 0.294i)10-s + (0.542 − 0.840i)11-s + (−0.797 + 0.603i)12-s + (0.995 + 0.0995i)13-s + (0.921 + 0.388i)14-s + (0.270 + 0.962i)15-s + (0.623 − 0.781i)16-s + (0.270 − 0.962i)17-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (0.980 − 0.198i)3-s + (−0.900 + 0.433i)4-s + (0.0747 + 0.997i)5-s + (−0.411 − 0.911i)6-s + (−0.583 + 0.811i)7-s + (0.623 + 0.781i)8-s + (0.921 − 0.388i)9-s + (0.955 − 0.294i)10-s + (0.542 − 0.840i)11-s + (−0.797 + 0.603i)12-s + (0.995 + 0.0995i)13-s + (0.921 + 0.388i)14-s + (0.270 + 0.962i)15-s + (0.623 − 0.781i)16-s + (0.270 − 0.962i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.857 - 0.515i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.857 - 0.515i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.167445562 - 0.3237648177i\)
\(L(\frac12)\) \(\approx\) \(1.167445562 - 0.3237648177i\)
\(L(1)\) \(\approx\) \(1.130015270 - 0.3123953491i\)
\(L(1)\) \(\approx\) \(1.130015270 - 0.3123953491i\)

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 \( 1 + (0.980 - 0.198i)T \)
5 \( 1 + (0.0747 + 0.997i)T \)
7 \( 1 + (-0.583 + 0.811i)T \)
11 \( 1 + (0.542 - 0.840i)T \)
13 \( 1 + (0.995 + 0.0995i)T \)
17 \( 1 + (0.270 - 0.962i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.542 + 0.840i)T \)
29 \( 1 + (-0.853 + 0.521i)T \)
31 \( 1 + (-0.969 - 0.246i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.969 + 0.246i)T \)
43 \( 1 + (-0.0249 - 0.999i)T \)
47 \( 1 + (0.826 + 0.563i)T \)
53 \( 1 + (-0.797 - 0.603i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.988 - 0.149i)T \)
67 \( 1 + (-0.661 - 0.749i)T \)
71 \( 1 + (-0.998 + 0.0498i)T \)
73 \( 1 + (-0.733 - 0.680i)T \)
79 \( 1 + (0.456 - 0.889i)T \)
83 \( 1 + (-0.124 + 0.992i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (0.878 - 0.478i)T \)
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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

−28.471192188230296776368040659390, −27.774141319196148596852312386328, −26.643704174640604079935055507915, −25.71564601254565468384255731344, −25.21927169311891270959973182570, −24.04183261202796556534142039849, −23.26050791075408836842268240970, −21.90580225118056010715857867590, −20.52973145519424506619848523960, −19.82574005133511799233725477426, −18.77050359910242336506696920648, −17.28952362896443473065707360637, −16.54069369932191765728542596953, −15.513988749058771311836177096021, −14.57659468026145816386543829288, −13.298009556814736136756826075001, −12.884902894107246393339179117140, −10.45320981365798325262821151849, −9.38057609360980146425835192347, −8.637859136427631562728942590419, −7.5278887146862788834965270646, −6.33846579933875619716049445892, −4.6222292659991448709728675500, −3.80913930570088801173074082700, −1.39640452529626783097914133609, 1.78063467438898707225083024917, 3.091665007910018225436510242866, 3.66658444343565289769204904142, 5.93083943608768876930829317629, 7.453422440516479407820006843406, 8.800869152772790593164892254523, 9.48370498322781847722047064078, 10.78397084146102764144988791391, 11.86960348547028821244917670923, 13.154571580695226951903441026900, 13.98667169433649026821237510561, 14.98913703579490349823619322596, 16.38956884067551408030786085940, 18.20199605674225461424894654755, 18.74480361053376886259270727882, 19.36157631403853308373552802324, 20.60489721160761263446554351065, 21.55808970427325403699865450287, 22.3400057745255126464390567423, 23.51036762551497067556596104843, 25.28855186163424275732174857959, 25.69931140954080431465979435589, 26.91032710063920422964281395221, 27.5395023701257278687191773312, 29.07888676562041612011267393504

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