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Properties

Label 1-127-127.47-r0-0-0
Degree $1$
Conductor $127$
Sign $0.999 + 0.0364i$
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 + (−0.988 + 0.149i)3-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)5-s + (0.0747 − 0.997i)6-s + (0.0747 − 0.997i)7-s + (0.623 − 0.781i)8-s + (0.955 − 0.294i)9-s + (−0.222 − 0.974i)10-s + (−0.733 + 0.680i)11-s + (0.955 + 0.294i)12-s + (0.0747 − 0.997i)13-s + (0.955 + 0.294i)14-s + (0.826 − 0.563i)15-s + (0.623 + 0.781i)16-s + (0.826 + 0.563i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.988 + 0.149i)3-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)5-s + (0.0747 − 0.997i)6-s + (0.0747 − 0.997i)7-s + (0.623 − 0.781i)8-s + (0.955 − 0.294i)9-s + (−0.222 − 0.974i)10-s + (−0.733 + 0.680i)11-s + (0.955 + 0.294i)12-s + (0.0747 − 0.997i)13-s + (0.955 + 0.294i)14-s + (0.826 − 0.563i)15-s + (0.623 + 0.781i)16-s + (0.826 + 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.999 + 0.0364i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.999 + 0.0364i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4615965102 + 0.008411142859i\)
\(L(\frac12)\) \(\approx\) \(0.4615965102 + 0.008411142859i\)
\(L(1)\) \(\approx\) \(0.5330842219 + 0.1551243507i\)
\(L(1)\) \(\approx\) \(0.5330842219 + 0.1551243507i\)

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

−28.80815417280143397321366311855, −27.93822496821497118560425296677, −27.34060253312561947052091553281, −26.19804822852846293946411973933, −24.47945720816260162076239014223, −23.61717873118704149610477155198, −22.68782072964181547335619361716, −21.630036879016691980233914115696, −20.92612728530328789177288364119, −19.42818227097326380809452740458, −18.67914190222111813716241780895, −17.92297674773935099824048005045, −16.447859001526150194068580889002, −15.84452569159253703535999148755, −13.94887356609515210244250351830, −12.61622678650107829183126567502, −11.82526061715275508720159757875, −11.323076236172722919673345052092, −9.90163212806135969243086280623, −8.65346591847160595311206039528, −7.508469791264808209367890713045, −5.58237382630282413883731428926, −4.64874364540054484777412617176, −3.11570578059968939367867929675, −1.27226480749238286245916535699, 0.6254848485470804770285116715, 3.799291665170769545879193912349, 4.85162031818426192749608330992, 6.109815586428178545224839076683, 7.39493476970311905696494483141, 7.90651111054073663031066102784, 10.06852243313711877626532608498, 10.523848553831459090709567599639, 12.06459568505747770322474493962, 13.248849772403285643000120935598, 14.6800637491281928497809385975, 15.657449006607410119824843222439, 16.38132546051477328402035833129, 17.537795141355686722128548190285, 18.2041907800389659669202690727, 19.411301076712564568272067280972, 20.73774430942991036456697754681, 22.45595373047115028449810697162, 22.97789499722721993361737210046, 23.62715266411888326650568053948, 24.59636875792041079430775962190, 26.178279720442690699828788095423, 26.72116557761759374923237548410, 27.74991931066815258250788890563, 28.38171202514241240546882794556

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