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Properties

Label 1-1480-1480.1027-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.997 - 0.0634i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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  + (−0.342 + 0.939i)3-s + (0.984 − 0.173i)7-s + (−0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (−0.642 − 0.766i)13-s + (0.642 − 0.766i)17-s + (−0.939 − 0.342i)19-s + (−0.173 + 0.984i)21-s + (−0.866 + 0.5i)23-s + (0.866 − 0.5i)27-s + (0.5 − 0.866i)29-s + 31-s + (−0.642 − 0.766i)33-s + (0.939 − 0.342i)39-s + (0.766 − 0.642i)41-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)3-s + (0.984 − 0.173i)7-s + (−0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (−0.642 − 0.766i)13-s + (0.642 − 0.766i)17-s + (−0.939 − 0.342i)19-s + (−0.173 + 0.984i)21-s + (−0.866 + 0.5i)23-s + (0.866 − 0.5i)27-s + (0.5 − 0.866i)29-s + 31-s + (−0.642 − 0.766i)33-s + (0.939 − 0.342i)39-s + (0.766 − 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.997 - 0.0634i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.997 - 0.0634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.247329518 - 0.03960480601i\)
\(L(\frac12)\) \(\approx\) \(1.247329518 - 0.03960480601i\)
\(L(1)\) \(\approx\) \(0.9540754776 + 0.1535881027i\)
\(L(1)\) \(\approx\) \(0.9540754776 + 0.1535881027i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.342 + 0.939i)T \)
7 \( 1 + (0.984 - 0.173i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + (0.642 - 0.766i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.984 + 0.173i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.642 - 0.766i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−20.81029980341337683606555634790, −19.561957180700835029673308046534, −19.222910474141518290065122155143, −18.29703117165099462015847534349, −17.88586853138196038451929647038, −16.7826234118326455391393763843, −16.5963132320468839346584608324, −15.26299901318107739774348142757, −14.35967884015780515997415794887, −13.986227130013512686246385256745, −12.972132297533040941780069535644, −12.20745891078863046611609874375, −11.6566364173340270586764325773, −10.79984413360759706311989794206, −10.131858329784192926436072437224, −8.57546103847899252391406899791, −8.34181446242416829081290773069, −7.45975854603066953466408168309, −6.521394616689693076703555473684, −5.77932405172074527949656527473, −4.99633387347823200740389081635, −4.01063742457112351419931979303, −2.64136714138803090416125497659, −1.94366541005629782850514988433, −0.963878217054549540250672393075, 0.59073636714912533871741314426, 2.09789690941858909117359130627, 2.965728473883945555182588016634, 4.19438241165599430899365570621, 4.77891188152245344276661207258, 5.4065691817570232354052640990, 6.40487004470667209781995984617, 7.63431846506937467529486632289, 8.10280587196472149058946608835, 9.24458576964191595150744896670, 10.07537957316679889445343976370, 10.472213141634394137464017392640, 11.51050460656179741205671345038, 12.02213103640240431275656148961, 12.98652873421928963975421270456, 14.10272621527831595941577132647, 14.71542652281653472045755108162, 15.40194703405603680948297309252, 15.998005194282584334894018880920, 17.0520273727860885449622397305, 17.61716237366926815310974718028, 18.02251094374396338650011535010, 19.30757593041762571120839697997, 20.11198423364651664907606608312, 20.84016882050987348023433285497

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