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Properties

Label 1-127-127.16-r0-0-0
Degree $1$
Conductor $127$
Sign $0.843 - 0.536i$
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 + (−0.222 + 0.974i)3-s + (0.623 + 0.781i)4-s + (0.623 − 0.781i)5-s + (0.623 − 0.781i)6-s + (0.623 − 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.900 − 0.433i)9-s + (−0.900 + 0.433i)10-s + (−0.900 + 0.433i)11-s + (−0.900 + 0.433i)12-s + (0.623 − 0.781i)13-s + (−0.900 + 0.433i)14-s + (0.623 + 0.781i)15-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (−0.222 + 0.974i)3-s + (0.623 + 0.781i)4-s + (0.623 − 0.781i)5-s + (0.623 − 0.781i)6-s + (0.623 − 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.900 − 0.433i)9-s + (−0.900 + 0.433i)10-s + (−0.900 + 0.433i)11-s + (−0.900 + 0.433i)12-s + (0.623 − 0.781i)13-s + (−0.900 + 0.433i)14-s + (0.623 + 0.781i)15-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.843 - 0.536i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.843 - 0.536i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7161954167 - 0.2086026635i\)
\(L(\frac12)\) \(\approx\) \(0.7161954167 - 0.2086026635i\)
\(L(1)\) \(\approx\) \(0.7545331239 - 0.1024609550i\)
\(L(1)\) \(\approx\) \(0.7545331239 - 0.1024609550i\)

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 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
7 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (0.623 + 0.781i)T \)
37 \( 1 + T \)
41 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (-0.900 - 0.433i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + (-0.900 - 0.433i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−28.63404291214558041872856798881, −28.262117079385303287867921929084, −26.64660860467917010021698291538, −25.88717496549326126590734612981, −25.01781925470459299572873202171, −24.10405626455219688781083837626, −23.30808017273200824185157124467, −21.82667377741467081257045477589, −20.71063400731547980087153673234, −19.13387813170609671909795401799, −18.51579732736230010335781397437, −17.962901160647210763740619852150, −16.91726225718974304775122903390, −15.59523209513738727466272103133, −14.40250229243103831795971314055, −13.53496252860245206003900706828, −11.7866848302945286360271149762, −11.033900502215509567150090804095, −9.7238126817612437530043358696, −8.32313554488783965050067085137, −7.542645797036783707886955379488, −6.12191061037370431299045875173, −5.64020001317113026688950867859, −2.614740213317425852144349629935, −1.59285271276600151013353733937, 1.07286880613261219776102703738, 2.93899005317935322932817725405, 4.48115903072328538256646346397, 5.67075419323534748505027646330, 7.59824809237301724236324025712, 8.6378414773680083996826743775, 9.87047717258787685808232644516, 10.42504449845914077343586589930, 11.59187437011335065545775135132, 12.86374608586848504437952735062, 14.216114727282674918343305459761, 15.90348396522713721684484838295, 16.382971735813329062329152225845, 17.64953225161518150964995335916, 18.06324061622503032888245674377, 20.10453541348715344561228604352, 20.58261390892030295650679803175, 21.16333798471514507598021794008, 22.43861347313895387558160220221, 23.75403352586963446306667256602, 25.10699977370177438335815345742, 25.98433405173436963744788145512, 26.93256604495308695772328602934, 27.79339816872378646172445043204, 28.51614809208026073149008610232

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