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Properties

Label 1-127-127.22-r0-0-0
Degree $1$
Conductor $127$
Sign $0.999 + 0.0435i$
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 + (0.766 + 0.642i)3-s + 4-s + (−0.5 − 0.866i)5-s + (0.766 + 0.642i)6-s + (0.173 − 0.984i)7-s + 8-s + (0.173 + 0.984i)9-s + (−0.5 − 0.866i)10-s + (−0.939 − 0.342i)11-s + (0.766 + 0.642i)12-s + (−0.939 + 0.342i)13-s + (0.173 − 0.984i)14-s + (0.173 − 0.984i)15-s + 16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + 2-s + (0.766 + 0.642i)3-s + 4-s + (−0.5 − 0.866i)5-s + (0.766 + 0.642i)6-s + (0.173 − 0.984i)7-s + 8-s + (0.173 + 0.984i)9-s + (−0.5 − 0.866i)10-s + (−0.939 − 0.342i)11-s + (0.766 + 0.642i)12-s + (−0.939 + 0.342i)13-s + (0.173 − 0.984i)14-s + (0.173 − 0.984i)15-s + 16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.999 + 0.0435i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.999 + 0.0435i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.135162492 + 0.04655331286i\)
\(L(\frac12)\) \(\approx\) \(2.135162492 + 0.04655331286i\)
\(L(1)\) \(\approx\) \(1.972546568 + 0.05822103657i\)
\(L(1)\) \(\approx\) \(1.972546568 + 0.05822103657i\)

Euler product

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

−29.30287858078394490689947056774, −27.956384236322165190974642148171, −26.4049965728978717953294372065, −25.67349545083155289617941208150, −24.636759464943419279728165399215, −23.88731217569625255122037989353, −22.78921653014039965915877843772, −21.88042907386985271737065607351, −20.754208836965060863068550640233, −19.75831341814128641028419104668, −18.80277144872022790720133375087, −17.87622354607378927825294641906, −15.85667885827886315146058595514, −15.0556820709058754734512214527, −14.442762950065960252243607130373, −13.17341697574036797285554582803, −12.25430668664106396100975424109, −11.32020036872475956555215398490, −9.793056505866109927156376660758, −7.99445804248637374516692803953, −7.27318321453745412750509083916, −6.02890474059252780531113038488, −4.535655927785801062993832539029, −2.78781363971515940215924517836, −2.46616611793290235061931999524, 2.01669080161564505175167886302, 3.695419010822330334748958062630, 4.36608951058355365167939636825, 5.52685768046687655909356744917, 7.505660064962752335432028652645, 8.197233297575761722046341894564, 9.95761140556838850531834987594, 10.89981212958689494239535425458, 12.35305563450987895875447729554, 13.327274287675178066174562347561, 14.2767899561034032489441392876, 15.26783920341359706782843046388, 16.2876891882105784493929927868, 16.96298259901675910586862978549, 19.31253230398306220289625308979, 19.93062095216930316664889801180, 20.93355509115930600343196609697, 21.44229492054765213493724729632, 22.874910762577646250365554787196, 23.893671446761986691912953057321, 24.51545195887434505515528510142, 25.84361218382821762050897857701, 26.675906358376370123467796425784, 27.82518371333972709008100421683, 28.97792847670813704239807846093

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