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Properties

Label 1-127-127.31-r0-0-0
Degree $1$
Conductor $127$
Sign $0.808 + 0.588i$
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 + (−0.661 + 0.749i)3-s + (−0.900 − 0.433i)4-s + (0.0747 − 0.997i)5-s + (−0.583 − 0.811i)6-s + (0.995 − 0.0995i)7-s + (0.623 − 0.781i)8-s + (−0.124 − 0.992i)9-s + (0.955 + 0.294i)10-s + (0.456 − 0.889i)11-s + (0.921 − 0.388i)12-s + (−0.411 + 0.911i)13-s + (−0.124 + 0.992i)14-s + (0.698 + 0.715i)15-s + (0.623 + 0.781i)16-s + (0.698 − 0.715i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.661 + 0.749i)3-s + (−0.900 − 0.433i)4-s + (0.0747 − 0.997i)5-s + (−0.583 − 0.811i)6-s + (0.995 − 0.0995i)7-s + (0.623 − 0.781i)8-s + (−0.124 − 0.992i)9-s + (0.955 + 0.294i)10-s + (0.456 − 0.889i)11-s + (0.921 − 0.388i)12-s + (−0.411 + 0.911i)13-s + (−0.124 + 0.992i)14-s + (0.698 + 0.715i)15-s + (0.623 + 0.781i)16-s + (0.698 − 0.715i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.808 + 0.588i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.808 + 0.588i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7400947572 + 0.2408978496i\)
\(L(\frac12)\) \(\approx\) \(0.7400947572 + 0.2408978496i\)
\(L(1)\) \(\approx\) \(0.7534082639 + 0.2794586799i\)
\(L(1)\) \(\approx\) \(0.7534082639 + 0.2794586799i\)

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 \( 1 + (-0.661 + 0.749i)T \)
5 \( 1 + (0.0747 - 0.997i)T \)
7 \( 1 + (0.995 - 0.0995i)T \)
11 \( 1 + (0.456 - 0.889i)T \)
13 \( 1 + (-0.411 + 0.911i)T \)
17 \( 1 + (0.698 - 0.715i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.456 + 0.889i)T \)
29 \( 1 + (-0.0249 + 0.999i)T \)
31 \( 1 + (0.270 - 0.962i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.270 + 0.962i)T \)
43 \( 1 + (0.878 - 0.478i)T \)
47 \( 1 + (0.826 - 0.563i)T \)
53 \( 1 + (0.921 + 0.388i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (-0.988 + 0.149i)T \)
67 \( 1 + (-0.318 - 0.947i)T \)
71 \( 1 + (0.542 - 0.840i)T \)
73 \( 1 + (-0.733 + 0.680i)T \)
79 \( 1 + (-0.998 - 0.0498i)T \)
83 \( 1 + (-0.797 + 0.603i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
97 \( 1 + (-0.853 + 0.521i)T \)
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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

−28.86342362931106325938067491605, −27.7254015848651853231864195307, −27.23218064122088856143876558712, −25.77952983174767739463686977500, −24.7528583326554256751000959878, −23.24373722629739350126973482051, −22.720869177633188013559535186774, −21.72259196632860807247270502611, −20.60334775281450637744434291151, −19.317088212397449073083121799670, −18.57362183661854281033955379147, −17.610872330207375561011064131, −17.13909367643662191012170749821, −14.90559093096965866123351346514, −14.06803074639488067900112638250, −12.61864624808517803258973638207, −11.94337024779124251933220191157, −10.782710029133950857573689586422, −10.15889912012089481641429711758, −8.24752133852967133271500514307, −7.359827543687036877847216057466, −5.79339916093719096630579604911, −4.3921924300451647233226670061, −2.61723055417983358404103044614, −1.49177576348078777949278832903, 1.014200739279626901085138719997, 4.125912647351577245947234106588, 4.9770292318162170893831479623, 5.86414269152331204069924847725, 7.38049562879377964047665177951, 8.834754139995362806695560707977, 9.39303679514099809278715963376, 10.95937539399927830372252909709, 12.03187778213176748089796394623, 13.62595413859166826417114076803, 14.62729106938470758592790056563, 15.765413504974546983147083404601, 16.77893193069886706231759368872, 17.12778366060138270881866312765, 18.32959098243215335719670407135, 19.755374829077445419872741051068, 21.21583523995362785946929700376, 21.76516456126616940570211989431, 23.244773812450437592038883991242, 23.99533846740949154933972566287, 24.68255427317911589680301714027, 26.03427093771071612317104172435, 27.19680837554055520471102786249, 27.5894150050930121863415467981, 28.531591926075865028174729917464

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