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Properties

Label 1-33e2-1089.1003-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.819 + 0.572i$
Analytic cond. $117.029$
Root an. cond. $117.029$
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  + (0.905 + 0.424i)2-s + (0.640 + 0.768i)4-s + (0.548 + 0.836i)5-s + (−0.997 + 0.0760i)7-s + (0.254 + 0.967i)8-s + (0.142 + 0.989i)10-s + (0.969 + 0.244i)13-s + (−0.935 − 0.353i)14-s + (−0.179 + 0.983i)16-s + (0.870 + 0.491i)17-s + (0.736 + 0.676i)19-s + (−0.290 + 0.956i)20-s + (0.235 + 0.971i)23-s + (−0.398 + 0.917i)25-s + (0.774 + 0.633i)26-s + ⋯
L(s)  = 1  + (0.905 + 0.424i)2-s + (0.640 + 0.768i)4-s + (0.548 + 0.836i)5-s + (−0.997 + 0.0760i)7-s + (0.254 + 0.967i)8-s + (0.142 + 0.989i)10-s + (0.969 + 0.244i)13-s + (−0.935 − 0.353i)14-s + (−0.179 + 0.983i)16-s + (0.870 + 0.491i)17-s + (0.736 + 0.676i)19-s + (−0.290 + 0.956i)20-s + (0.235 + 0.971i)23-s + (−0.398 + 0.917i)25-s + (0.774 + 0.633i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.819 + 0.572i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.819 + 0.572i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.313886032 + 4.176565021i\)
\(L(\frac12)\) \(\approx\) \(1.313886032 + 4.176565021i\)
\(L(1)\) \(\approx\) \(1.630774979 + 1.201701593i\)
\(L(1)\) \(\approx\) \(1.630774979 + 1.201701593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.905 + 0.424i)T \)
5 \( 1 + (0.548 + 0.836i)T \)
7 \( 1 + (-0.997 + 0.0760i)T \)
13 \( 1 + (0.969 + 0.244i)T \)
17 \( 1 + (0.870 + 0.491i)T \)
19 \( 1 + (0.736 + 0.676i)T \)
23 \( 1 + (0.235 + 0.971i)T \)
29 \( 1 + (0.595 - 0.803i)T \)
31 \( 1 + (0.999 - 0.0380i)T \)
37 \( 1 + (0.897 - 0.441i)T \)
41 \( 1 + (-0.797 - 0.603i)T \)
43 \( 1 + (-0.0475 - 0.998i)T \)
47 \( 1 + (-0.999 + 0.0190i)T \)
53 \( 1 + (0.941 + 0.336i)T \)
59 \( 1 + (0.123 + 0.992i)T \)
61 \( 1 + (-0.820 - 0.572i)T \)
67 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (0.198 + 0.980i)T \)
73 \( 1 + (0.0285 + 0.999i)T \)
79 \( 1 + (-0.988 - 0.151i)T \)
83 \( 1 + (-0.380 - 0.924i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.548 - 0.836i)T \)
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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

−21.0416884957073169730930534641, −20.18284563856222942867418349318, −19.78339711891210741394494803278, −18.695428697449540498839021292130, −18.00931539953289598803000229125, −16.61730048709807220094537549699, −16.27454710490135060293499861352, −15.50093487361097632652417822582, −14.440771058705718251510716674301, −13.52704719185362631093508665428, −13.20207557847800990221534373508, −12.36460418489167587998011677150, −11.674013309336485525449935644390, −10.55793408295994032484279955546, −9.84989395930431722436420224445, −9.15817969600168848399387201603, −8.03665403349492984249044816709, −6.68665067332643369304582836, −6.17853457883498006544801637553, −5.20364465012832640411079239760, −4.53243377304958363611755325963, −3.31689684923302978384564533592, −2.75191296654685659033945988145, −1.3322327417825584514737849698, −0.65853677995131967054409924871, 1.39304095716701523745816819487, 2.62944313714691234613664684724, 3.37249419307129025714448376581, 4.02171527454476075234216476962, 5.5682322131184751681606943440, 5.95083596675761862466128614268, 6.76094741467835536244485909467, 7.535175442319824451813171768978, 8.56369417327626180298331219245, 9.760182863448560968027860705586, 10.394522459687046939594958961573, 11.478113192246962484566283182430, 12.14040831591290971046529964869, 13.22997434386757480581637420700, 13.65285259377839034011683471887, 14.40187570656851053069588228479, 15.31981867005713177873563482319, 15.90469632319303305149267137735, 16.74685060240588905994425332911, 17.49769612458666460590816977250, 18.51428192858271414780043489991, 19.166177701165812881061998572807, 20.15970970601591323971947121122, 21.1896145024578333764187785614, 21.53945487670455594395548232939

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