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Properties

Label 1-33e2-1089.272-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.956 - 0.292i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (−0.595 − 0.803i)2-s + (−0.290 + 0.956i)4-s + (−0.997 + 0.0760i)5-s + (0.969 + 0.244i)7-s + (0.941 − 0.336i)8-s + (0.654 + 0.755i)10-s + (0.999 − 0.0190i)13-s + (−0.380 − 0.924i)14-s + (−0.830 − 0.556i)16-s + (0.774 + 0.633i)17-s + (0.998 − 0.0570i)19-s + (0.217 − 0.976i)20-s + (−0.928 − 0.371i)23-s + (0.988 − 0.151i)25-s + (−0.610 − 0.791i)26-s + ⋯
L(s)  = 1  + (−0.595 − 0.803i)2-s + (−0.290 + 0.956i)4-s + (−0.997 + 0.0760i)5-s + (0.969 + 0.244i)7-s + (0.941 − 0.336i)8-s + (0.654 + 0.755i)10-s + (0.999 − 0.0190i)13-s + (−0.380 − 0.924i)14-s + (−0.830 − 0.556i)16-s + (0.774 + 0.633i)17-s + (0.998 − 0.0570i)19-s + (0.217 − 0.976i)20-s + (−0.928 − 0.371i)23-s + (0.988 − 0.151i)25-s + (−0.610 − 0.791i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.956 - 0.292i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.956 - 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.047827084 - 0.1565215495i\)
\(L(\frac12)\) \(\approx\) \(1.047827084 - 0.1565215495i\)
\(L(1)\) \(\approx\) \(0.7948001090 - 0.1742321376i\)
\(L(1)\) \(\approx\) \(0.7948001090 - 0.1742321376i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.595 - 0.803i)T \)
5 \( 1 + (-0.997 + 0.0760i)T \)
7 \( 1 + (0.969 + 0.244i)T \)
13 \( 1 + (0.999 - 0.0190i)T \)
17 \( 1 + (0.774 + 0.633i)T \)
19 \( 1 + (0.998 - 0.0570i)T \)
23 \( 1 + (-0.928 - 0.371i)T \)
29 \( 1 + (-0.625 + 0.780i)T \)
31 \( 1 + (0.123 - 0.992i)T \)
37 \( 1 + (0.0855 + 0.996i)T \)
41 \( 1 + (0.861 - 0.508i)T \)
43 \( 1 + (-0.235 - 0.971i)T \)
47 \( 1 + (-0.749 + 0.662i)T \)
53 \( 1 + (-0.897 - 0.441i)T \)
59 \( 1 + (-0.00951 + 0.999i)T \)
61 \( 1 + (0.398 + 0.917i)T \)
67 \( 1 + (-0.995 - 0.0950i)T \)
71 \( 1 + (0.254 - 0.967i)T \)
73 \( 1 + (0.466 - 0.884i)T \)
79 \( 1 + (-0.879 + 0.475i)T \)
83 \( 1 + (0.640 + 0.768i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.997 + 0.0760i)T \)
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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

−21.342606146685357562238741159892, −20.43611026876382970651974384191, −19.901769933123892510754172139249, −18.94724274646270663712042480013, −18.24244504132181836253764146971, −17.721625452465795957914617984884, −16.64052015065465897462817642333, −15.99914851313832955356792165038, −15.50697233675495812018134967422, −14.36298817136709184581851783926, −14.107022567575774739963959224767, −12.90079926144582843718672011125, −11.56584582900865563757791118591, −11.298899060186762960675786167685, −10.220851643378806528579268257574, −9.30418151394112261832843422127, −8.33269165079456542720418406021, −7.80168652139552491207999793498, −7.223869804653890537343720230963, −6.0431099794167597153732053359, −5.16587241624061008722205407097, −4.33461020564259075038016189941, −3.35900496561892278151706093027, −1.67677790989722990892291801450, −0.773365450356118865765316569789, 0.94307303925217076296013286133, 1.83217662305560974515692660186, 3.10261595587412240650243329325, 3.84110663228919370806405992322, 4.65817353221250611345728362945, 5.834489860712570048453547177694, 7.22971989357145198542824004413, 7.98882843697466051080657486055, 8.415160262934005944280264097106, 9.38102839859392241816295007795, 10.47720267237738602236371541560, 11.10795010776773862288996296365, 11.801301700557281596756577190409, 12.3439491158974279319216229421, 13.40043341077102871323410676955, 14.31176061395832509269303944428, 15.21502677137207839959051051496, 16.12526876298492968470279832855, 16.75665387766051211112008737384, 17.84632216648071050158332405855, 18.39778049121543567665800673087, 19.00111704244476156258770652916, 19.892844042840953285217470789323, 20.64344136086419774877046601394, 21.01999768934366116660123941523

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