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Properties

Label 1-1176-1176.5-r0-0-0
Degree $1$
Conductor $1176$
Sign $0.414 - 0.910i$
Analytic cond. $5.46132$
Root an. cond. $5.46132$
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  + (0.988 + 0.149i)5-s + (−0.733 − 0.680i)11-s + (−0.222 + 0.974i)13-s + (0.0747 − 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.900 + 0.433i)29-s + (0.5 − 0.866i)31-s + (−0.826 − 0.563i)37-s + (0.623 − 0.781i)41-s + (−0.623 − 0.781i)43-s + (0.955 − 0.294i)47-s + (0.826 − 0.563i)53-s + (−0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)5-s + (−0.733 − 0.680i)11-s + (−0.222 + 0.974i)13-s + (0.0747 − 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.900 + 0.433i)29-s + (0.5 − 0.866i)31-s + (−0.826 − 0.563i)37-s + (0.623 − 0.781i)41-s + (−0.623 − 0.781i)43-s + (0.955 − 0.294i)47-s + (0.826 − 0.563i)53-s + (−0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.414 - 0.910i$
Analytic conductor: \(5.46132\)
Root analytic conductor: \(5.46132\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1176,\ (0:\ ),\ 0.414 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.264209311 - 0.8133259135i\)
\(L(\frac12)\) \(\approx\) \(1.264209311 - 0.8133259135i\)
\(L(1)\) \(\approx\) \(1.135571428 - 0.1774547234i\)
\(L(1)\) \(\approx\) \(1.135571428 - 0.1774547234i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.988 + 0.149i)T \)
11 \( 1 + (-0.733 - 0.680i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (0.0747 - 0.997i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.0747 - 0.997i)T \)
29 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.826 - 0.563i)T \)
41 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (0.826 - 0.563i)T \)
59 \( 1 + (0.988 - 0.149i)T \)
61 \( 1 + (0.826 + 0.563i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
97 \( 1 - T \)
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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

−21.30470130099446309556119728154, −20.72202002653154847838823803597, −19.95371867412954794352172349831, −19.0234209850064949265135619193, −18.1646838153236749877315222782, −17.469790412882226437676172735642, −17.02133232467465974333464595407, −15.923022969185228179340117946933, −15.10867941035492414480869966179, −14.468777914132189877377722717116, −13.41111744649892574610203489271, −12.88118166585631704032862659107, −12.21793393640209592530229688103, −10.956960406781607021106064120348, −10.12720781783674607642790147473, −9.80652972846298057055764156857, −8.57647536717954705953245889110, −7.88814293548079566748625214691, −6.89312682506835096100424061421, −5.81374583094324829560173101569, −5.382883289168155550195159992392, −4.28380234523200726717944505475, −3.13305387390362311334657844161, −2.15795290832409770593758520943, −1.31385542071347893515267430473, 0.60064123632972165059536074508, 2.13792693998975981373982159315, 2.58214664977666774412630924013, 3.87446150715776295718294435646, 4.99840870174049068812380133216, 5.634353852289420983580361044155, 6.667349834351163387606913828326, 7.26332334813903630573465951649, 8.58563588827586330453110411753, 9.14284173460181417327356659972, 10.04845325229064743242740270443, 10.812007028915199392303386163713, 11.57283315803271504265764320104, 12.629093316425183475562424055831, 13.455851531909259109045154182172, 13.96865537536649253884580777343, 14.76438470560770650639690701363, 15.76877584150842854207471650511, 16.57727449409307698377668043616, 17.17481218752520883745495123347, 18.15128821301067133766470631404, 18.67347434844095762765340907837, 19.41369261884816301357129786069, 20.71533898879463431879420885336, 20.9209724105751950767327429181

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