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Properties

Label 1-127-127.62-r0-0-0
Degree $1$
Conductor $127$
Sign $0.415 - 0.909i$
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  + (−0.900 + 0.433i)2-s + (0.921 − 0.388i)3-s + (0.623 − 0.781i)4-s + (−0.988 + 0.149i)5-s + (−0.661 + 0.749i)6-s + (−0.318 − 0.947i)7-s + (−0.222 + 0.974i)8-s + (0.698 − 0.715i)9-s + (0.826 − 0.563i)10-s + (−0.411 − 0.911i)11-s + (0.270 − 0.962i)12-s + (0.980 + 0.198i)13-s + (0.698 + 0.715i)14-s + (−0.853 + 0.521i)15-s + (−0.222 − 0.974i)16-s + (−0.853 − 0.521i)17-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.921 − 0.388i)3-s + (0.623 − 0.781i)4-s + (−0.988 + 0.149i)5-s + (−0.661 + 0.749i)6-s + (−0.318 − 0.947i)7-s + (−0.222 + 0.974i)8-s + (0.698 − 0.715i)9-s + (0.826 − 0.563i)10-s + (−0.411 − 0.911i)11-s + (0.270 − 0.962i)12-s + (0.980 + 0.198i)13-s + (0.698 + 0.715i)14-s + (−0.853 + 0.521i)15-s + (−0.222 − 0.974i)16-s + (−0.853 − 0.521i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.415 - 0.909i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.415 - 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6188731543 - 0.3978670923i\)
\(L(\frac12)\) \(\approx\) \(0.6188731543 - 0.3978670923i\)
\(L(1)\) \(\approx\) \(0.7591101350 - 0.1673713573i\)
\(L(1)\) \(\approx\) \(0.7591101350 - 0.1673713573i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad127 \( 1 \)
good2 \( 1 + (-0.900 + 0.433i)T \)
3 \( 1 + (0.921 - 0.388i)T \)
5 \( 1 + (-0.988 + 0.149i)T \)
7 \( 1 + (-0.318 - 0.947i)T \)
11 \( 1 + (-0.411 - 0.911i)T \)
13 \( 1 + (0.980 + 0.198i)T \)
17 \( 1 + (-0.853 - 0.521i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.411 + 0.911i)T \)
29 \( 1 + (0.456 - 0.889i)T \)
31 \( 1 + (0.878 + 0.478i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.878 - 0.478i)T \)
43 \( 1 + (-0.998 + 0.0498i)T \)
47 \( 1 + (0.365 + 0.930i)T \)
53 \( 1 + (0.270 + 0.962i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.955 + 0.294i)T \)
67 \( 1 + (-0.124 + 0.992i)T \)
71 \( 1 + (0.995 - 0.0995i)T \)
73 \( 1 + (0.0747 + 0.997i)T \)
79 \( 1 + (-0.583 - 0.811i)T \)
83 \( 1 + (-0.969 - 0.246i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
97 \( 1 + (0.542 - 0.840i)T \)
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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

−28.50544838123733517189876754064, −28.02862385696136120324399134514, −27.06441004464613341342134522140, −26.07829073632388380045350135795, −25.3935042499399365373230525475, −24.382256460171036238977111028836, −22.79947573103377934480114050268, −21.581802260295925232423748484676, −20.56069464170162612464254389380, −19.909981218041273216947490639689, −18.88658137606269710415793879035, −18.20887694359637434518826474687, −16.4784688205812524215987720870, −15.60495598631365188085434399393, −15.01094965093631946166839372798, −13.02705491992012994130905879668, −12.219266119367674239571488765639, −10.85874664060425625417832680121, −9.81933125264673133169909696185, −8.49710533898879503333143169481, −8.21240373656736933699211781378, −6.65891232448323210547653209606, −4.38866776802933857370216813783, −3.22089795652773049220475457666, −1.99452227633004037862958525132, 0.84090234052337298140982073798, 2.78382473272142851125839193756, 4.14030321567365941469178894856, 6.39045385848413931757626467200, 7.33658227364265372764217012344, 8.22737775576919455836504406151, 9.12124651541949505896886708360, 10.5753893200969891841937610307, 11.5051175462261017982463085113, 13.309570607180880517245164948085, 14.144736834068700703547692426200, 15.64242354016392683334705747915, 15.941424727609940111315889274107, 17.5143576208218859159017991742, 18.63783102019949665326828736167, 19.46476358194799061958948707825, 20.02462181531663958471949022535, 21.14025472514991697959378389206, 23.19451477505623321040428262128, 23.79442128065633210468229942468, 24.710193465459574578049930807958, 26.021215350490332343252908471633, 26.49255033207031542260101398102, 27.242919445924208860435824611, 28.52118378685378373518043207773

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