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Properties

Label 1-127-127.13-r0-0-0
Degree $1$
Conductor $127$
Sign $-0.971 - 0.237i$
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 + (−0.0249 − 0.999i)3-s + (−0.900 + 0.433i)4-s + (0.826 − 0.563i)5-s + (−0.969 + 0.246i)6-s + (0.270 − 0.962i)7-s + (0.623 + 0.781i)8-s + (−0.998 + 0.0498i)9-s + (−0.733 − 0.680i)10-s + (−0.124 − 0.992i)11-s + (0.456 + 0.889i)12-s + (0.698 + 0.715i)13-s + (−0.998 − 0.0498i)14-s + (−0.583 − 0.811i)15-s + (0.623 − 0.781i)16-s + (−0.583 + 0.811i)17-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.0249 − 0.999i)3-s + (−0.900 + 0.433i)4-s + (0.826 − 0.563i)5-s + (−0.969 + 0.246i)6-s + (0.270 − 0.962i)7-s + (0.623 + 0.781i)8-s + (−0.998 + 0.0498i)9-s + (−0.733 − 0.680i)10-s + (−0.124 − 0.992i)11-s + (0.456 + 0.889i)12-s + (0.698 + 0.715i)13-s + (−0.998 − 0.0498i)14-s + (−0.583 − 0.811i)15-s + (0.623 − 0.781i)16-s + (−0.583 + 0.811i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $-0.971 - 0.237i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ -0.971 - 0.237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1129773733 - 0.9387876466i\)
\(L(\frac12)\) \(\approx\) \(0.1129773733 - 0.9387876466i\)
\(L(1)\) \(\approx\) \(0.5514887427 - 0.7775102744i\)
\(L(1)\) \(\approx\) \(0.5514887427 - 0.7775102744i\)

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 \( 1 + (-0.0249 - 0.999i)T \)
5 \( 1 + (0.826 - 0.563i)T \)
7 \( 1 + (0.270 - 0.962i)T \)
11 \( 1 + (-0.124 - 0.992i)T \)
13 \( 1 + (0.698 + 0.715i)T \)
17 \( 1 + (-0.583 + 0.811i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.124 + 0.992i)T \)
29 \( 1 + (-0.318 + 0.947i)T \)
31 \( 1 + (-0.411 + 0.911i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (-0.411 - 0.911i)T \)
43 \( 1 + (0.980 - 0.198i)T \)
47 \( 1 + (0.0747 - 0.997i)T \)
53 \( 1 + (0.456 - 0.889i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (0.365 + 0.930i)T \)
67 \( 1 + (0.878 + 0.478i)T \)
71 \( 1 + (0.921 - 0.388i)T \)
73 \( 1 + (0.955 - 0.294i)T \)
79 \( 1 + (-0.797 - 0.603i)T \)
83 \( 1 + (0.542 + 0.840i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
97 \( 1 + (-0.661 + 0.749i)T \)
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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

−28.81412506176119128502618995244, −28.06721458780584541739430327657, −27.19556839520976904543118728213, −26.125347490803315611348259613700, −25.41043543441202797082909012109, −24.71769482541804277800087398291, −22.886651936118673483521747975865, −22.536811814495002461407917711531, −21.39751063505690500737958983414, −20.383474726652342132414341083963, −18.59243803193400146315149492768, −17.93691971145738125786476015554, −16.94403989167487110968416554696, −15.707141194669968541606746079199, −15.004590753844921992020446996758, −14.22794876257638317449949081013, −12.841259723493693052657160922804, −11.02847981958639998541451696188, −9.92080344828481190558243296677, −9.17180175463222840770961744845, −7.963278640433083974503431434181, −6.27537581744619500275187954379, −5.51793427376406649823099830403, −4.29991018339005506012569384330, −2.441819012503912798365081618659, 1.034558739733442661693431891292, 2.040470892981489310409702168229, 3.71370348257756979679959782325, 5.30243917932154641893214134960, 6.74666999728618277421372752501, 8.305306799805384513249229977381, 9.04073759764314460385621430206, 10.6455446905621873300512939983, 11.410289630396549025960028079158, 12.85192488107445939563725944688, 13.49504839911863634732313678621, 14.13696776379928224456418651503, 16.58072760249744529166474411810, 17.370624496527332732413085615921, 18.16554861181447935437434308150, 19.32863044085619549398648108210, 20.09819987825220421750410843045, 21.17655767095657112423544186120, 22.02471086456107147755752095284, 23.6782216601331913960121604030, 23.937013778366280093283004535275, 25.598773423670196368725240236181, 26.29263537600422081722104982489, 27.65455411881835546616949307629, 28.77006972232387021889728755590

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