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Properties

Label 1-127-127.68-r0-0-0
Degree $1$
Conductor $127$
Sign $0.999 + 0.0366i$
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  + 2-s + (−0.939 + 0.342i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.939 + 0.342i)6-s + (0.766 + 0.642i)7-s + 8-s + (0.766 − 0.642i)9-s + (−0.5 − 0.866i)10-s + (0.173 + 0.984i)11-s + (−0.939 + 0.342i)12-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)14-s + (0.766 + 0.642i)15-s + 16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.939 + 0.342i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.939 + 0.342i)6-s + (0.766 + 0.642i)7-s + 8-s + (0.766 − 0.642i)9-s + (−0.5 − 0.866i)10-s + (0.173 + 0.984i)11-s + (−0.939 + 0.342i)12-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)14-s + (0.766 + 0.642i)15-s + 16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.999 + 0.0366i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.999 + 0.0366i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.513840057 + 0.02773873561i\)
\(L(\frac12)\) \(\approx\) \(1.513840057 + 0.02773873561i\)
\(L(1)\) \(\approx\) \(1.446621188 + 0.02874358820i\)
\(L(1)\) \(\approx\) \(1.446621188 + 0.02874358820i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad127 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (0.173 + 0.984i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−29.18981807348860794754646997765, −27.95113278625254389081875064261, −26.88540338524517290500842306561, −25.71144344770815008009023421468, −24.10915465180964253360124244037, −23.79743642356391353665923513467, −22.961777042318361259621233423252, −21.80517384004585248707503183532, −21.26208768857805572675509890248, −19.55999308147768240589721202051, −18.79871201422833752439842829281, −17.31617978849308014706023671852, −16.453003169387816625565921291477, −15.26172043065084177730213046716, −14.144117322183136118284252024189, −13.29593812075725638006159471496, −11.722641152031219428441996393815, −11.31691675588500905313849271772, −10.42035091167863535187905005058, −7.91394923264614481933845262908, −6.931359392187938241193135904874, −6.01039919084744896585264265389, −4.63712319194395846610080199402, −3.52596043960935600302920864866, −1.6500968606300151659919396905, 1.5991558027787789051233512354, 3.69095243357360821283042987717, 4.989485849256829655263978404460, 5.37911741627682264727716991570, 6.97930720679228959039698474609, 8.32573820745185601180766252120, 10.09865738622736112072278092983, 11.26825933501849365989529736587, 12.36776657145020349780517616942, 12.58595271629731384513608483215, 14.54028206024824973783368291528, 15.38574801488837140039927840240, 16.30103424008336905553120890783, 17.259204027506622504875757291282, 18.535241624053428353899848262100, 20.31464858812259173515250766871, 20.780027975365979289344538124058, 21.891474386770026635526578359093, 22.94767027993243808642674903866, 23.52141843346928089621923524914, 24.64254816937944995567970415969, 25.32921707458817744136819174386, 27.348889317502693626869776421446, 27.90510947704405252056403708081, 28.79358521251148561375605036430

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