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Properties

Label 1-1176-1176.101-r0-0-0
Degree $1$
Conductor $1176$
Sign $0.580 + 0.814i$
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  + (−0.365 − 0.930i)5-s + (0.955 − 0.294i)11-s + (−0.222 + 0.974i)13-s + (0.826 + 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.826 + 0.563i)23-s + (−0.733 + 0.680i)25-s + (−0.900 + 0.433i)29-s + (0.5 + 0.866i)31-s + (−0.0747 + 0.997i)37-s + (0.623 − 0.781i)41-s + (−0.623 − 0.781i)43-s + (−0.733 − 0.680i)47-s + (0.0747 + 0.997i)53-s + (−0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)5-s + (0.955 − 0.294i)11-s + (−0.222 + 0.974i)13-s + (0.826 + 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.826 + 0.563i)23-s + (−0.733 + 0.680i)25-s + (−0.900 + 0.433i)29-s + (0.5 + 0.866i)31-s + (−0.0747 + 0.997i)37-s + (0.623 − 0.781i)41-s + (−0.623 − 0.781i)43-s + (−0.733 − 0.680i)47-s + (0.0747 + 0.997i)53-s + (−0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.580 + 0.814i$
Analytic conductor: \(5.46132\)
Root analytic conductor: \(5.46132\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1176,\ (0:\ ),\ 0.580 + 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.051764310 + 0.5415757202i\)
\(L(\frac12)\) \(\approx\) \(1.051764310 + 0.5415757202i\)
\(L(1)\) \(\approx\) \(0.9812963300 + 0.04422289761i\)
\(L(1)\) \(\approx\) \(0.9812963300 + 0.04422289761i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.365 - 0.930i)T \)
11 \( 1 + (0.955 - 0.294i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (0.826 + 0.563i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.826 + 0.563i)T \)
29 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.0747 + 0.997i)T \)
41 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (-0.733 - 0.680i)T \)
53 \( 1 + (0.0747 + 0.997i)T \)
59 \( 1 + (-0.365 + 0.930i)T \)
61 \( 1 + (0.0747 - 0.997i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (0.733 - 0.680i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
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

−21.18233965270628110807859668767, −20.1654884911132455583773294946, −19.61366077624890863440551860762, −18.84506298910374101120730410520, −18.03197184195012359221340803543, −17.41308690643693646352897666752, −16.46959499830576731553083784463, −15.601289070132717696217152316455, −14.764643185480486637261366111175, −14.421671111407477414417766717432, −13.332730042397786021566841132133, −12.44852314822365974457769648895, −11.59890791613670664097495638848, −10.9968036562406253450296357899, −9.996492272449554898414417951570, −9.43045509033262362426166082507, −8.13684251809521911497515476343, −7.5567925657454362196552041523, −6.61969887552542048967107377734, −5.934267884365483478242044416252, −4.72961457222218967378688981404, −3.81212373121530996866713682974, −2.958575532773666089666483876, −2.04644288718843462373005167016, −0.51603132435483216332356345556, 1.21353001877399044748268391510, 1.90425095872343535868016938706, 3.58790215421627028547226730528, 4.02583391480314420064052579493, 5.100079440728595899264585492, 5.96544872113008614793699273321, 6.88774133491339514235330603274, 7.93176115420998639621592590897, 8.61916574930658280280445877033, 9.38203010937635800139880562358, 10.189465876506229684078286260161, 11.34620194654304821443500953002, 12.08140661489928343099837586973, 12.50377745640841410983406358334, 13.67177016064991214329275561122, 14.29146714041617490120944063097, 15.169000211503435611334772519844, 16.11144222670955615673621528112, 16.87214524468543600693661228483, 17.09378830000030468388540010404, 18.42773044649303326693133937193, 19.19776615449294481416860328364, 19.74072049780986875622397938452, 20.57675135444885987963988003539, 21.365726864324473092495949281593

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