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Properties

Label 1-127-127.76-r0-0-0
Degree $1$
Conductor $127$
Sign $0.809 + 0.586i$
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 + (0.365 − 0.930i)3-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)5-s + (0.826 + 0.563i)6-s + (0.826 + 0.563i)7-s + (0.623 − 0.781i)8-s + (−0.733 − 0.680i)9-s + (−0.222 − 0.974i)10-s + (0.955 + 0.294i)11-s + (−0.733 + 0.680i)12-s + (0.826 + 0.563i)13-s + (−0.733 + 0.680i)14-s + (0.0747 + 0.997i)15-s + (0.623 + 0.781i)16-s + (0.0747 − 0.997i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (0.365 − 0.930i)3-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)5-s + (0.826 + 0.563i)6-s + (0.826 + 0.563i)7-s + (0.623 − 0.781i)8-s + (−0.733 − 0.680i)9-s + (−0.222 − 0.974i)10-s + (0.955 + 0.294i)11-s + (−0.733 + 0.680i)12-s + (0.826 + 0.563i)13-s + (−0.733 + 0.680i)14-s + (0.0747 + 0.997i)15-s + (0.623 + 0.781i)16-s + (0.0747 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.809 + 0.586i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.809 + 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9252127032 + 0.2999316090i\)
\(L(\frac12)\) \(\approx\) \(0.9252127032 + 0.2999316090i\)
\(L(1)\) \(\approx\) \(0.9390907834 + 0.2375480782i\)
\(L(1)\) \(\approx\) \(0.9390907834 + 0.2375480782i\)

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 \( 1 + (0.365 - 0.930i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (0.826 + 0.563i)T \)
11 \( 1 + (0.955 + 0.294i)T \)
13 \( 1 + (0.826 + 0.563i)T \)
17 \( 1 + (0.0747 - 0.997i)T \)
19 \( 1 + T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (-0.988 + 0.149i)T \)
31 \( 1 + (0.0747 + 0.997i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.0747 - 0.997i)T \)
43 \( 1 + (-0.988 + 0.149i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (-0.733 - 0.680i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 + (0.365 - 0.930i)T \)
71 \( 1 + (0.955 - 0.294i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 + (0.955 - 0.294i)T \)
83 \( 1 + (-0.733 - 0.680i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (-0.988 - 0.149i)T \)
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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

−28.23542489209420530376222230403, −27.8291621057289559694042071277, −26.970013299197482058893375405492, −26.30041944514814153907196593729, −24.76800175450976199920349167707, −23.385107379934777873097868279988, −22.49473399641929360035238222282, −21.35744170071044154606448901510, −20.46786152362046709453179078143, −19.937346803075187535428689263411, −18.89196232267391194188521884490, −17.35168386581042301907670360424, −16.574471666914097399664067703343, −15.19263540308157117323460279863, −14.16160045669695967607667512545, −12.96969345426964526817399783937, −11.37018706399996606494209942535, −11.090924824623163159801441629572, −9.64418318107334764848786987041, −8.5638356883982440503231431709, −7.82126434304468541298534872097, −5.2270395757118629578613080072, −4.038633157225393818761504938105, −3.43368573978004797451652929957, −1.31266566161600113166041114425, 1.38130921615481620453144909448, 3.45312231246580669396437798089, 5.03392273153663777505979792609, 6.59585029355054679205691150215, 7.31801039163608813428397270920, 8.40631323201749823578575216022, 9.20664345778480887682937656740, 11.27696203741815766507434974433, 12.14556948953401276427061826381, 13.71177814880438524460341618138, 14.50742913645326463226969251828, 15.32522005131016702305220588662, 16.57798856547883666107427289110, 17.92994059825766458643616048545, 18.52086242767458245289472854675, 19.3758791924516601009885854850, 20.59207501883705677417805024346, 22.38011827272750245351490877684, 23.162114770292406452751486810096, 24.15945136992424423778650574843, 24.8102260312612432795748196233, 25.74745632280530466445431009921, 26.85575962374391242003355682220, 27.637961870660360195024005438514, 28.73333873639754329546651470804

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