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Properties

Label 1-33e2-1089.299-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.484 + 0.874i$
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  + (−0.179 − 0.983i)2-s + (−0.935 + 0.353i)4-s + (0.683 − 0.730i)5-s + (−0.953 − 0.299i)7-s + (0.516 + 0.856i)8-s + (−0.841 − 0.540i)10-s + (−0.548 + 0.836i)13-s + (−0.123 + 0.992i)14-s + (0.749 − 0.662i)16-s + (−0.466 − 0.884i)17-s + (0.985 + 0.170i)19-s + (−0.380 + 0.924i)20-s + (−0.580 − 0.814i)23-s + (−0.0665 − 0.997i)25-s + (0.921 + 0.389i)26-s + ⋯
L(s)  = 1  + (−0.179 − 0.983i)2-s + (−0.935 + 0.353i)4-s + (0.683 − 0.730i)5-s + (−0.953 − 0.299i)7-s + (0.516 + 0.856i)8-s + (−0.841 − 0.540i)10-s + (−0.548 + 0.836i)13-s + (−0.123 + 0.992i)14-s + (0.749 − 0.662i)16-s + (−0.466 − 0.884i)17-s + (0.985 + 0.170i)19-s + (−0.380 + 0.924i)20-s + (−0.580 − 0.814i)23-s + (−0.0665 − 0.997i)25-s + (0.921 + 0.389i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.484 + 0.874i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.484 + 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1709500496 - 0.2900720603i\)
\(L(\frac12)\) \(\approx\) \(-0.1709500496 - 0.2900720603i\)
\(L(1)\) \(\approx\) \(0.5591876439 - 0.4394916314i\)
\(L(1)\) \(\approx\) \(0.5591876439 - 0.4394916314i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.179 - 0.983i)T \)
5 \( 1 + (0.683 - 0.730i)T \)
7 \( 1 + (-0.953 - 0.299i)T \)
13 \( 1 + (-0.548 + 0.836i)T \)
17 \( 1 + (-0.466 - 0.884i)T \)
19 \( 1 + (0.985 + 0.170i)T \)
23 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (-0.830 - 0.556i)T \)
31 \( 1 + (0.988 + 0.151i)T \)
37 \( 1 + (-0.254 + 0.967i)T \)
41 \( 1 + (-0.851 - 0.524i)T \)
43 \( 1 + (-0.981 - 0.189i)T \)
47 \( 1 + (-0.997 - 0.0760i)T \)
53 \( 1 + (-0.198 + 0.980i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (0.761 + 0.647i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (-0.696 - 0.717i)T \)
73 \( 1 + (-0.993 - 0.113i)T \)
79 \( 1 + (-0.820 + 0.572i)T \)
83 \( 1 + (0.00951 + 0.999i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (-0.683 - 0.730i)T \)
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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

−22.016081136068343887620561094789, −21.706779295476312972063172568499, −20.18997441739019822510413686369, −19.40010892464839057386364013630, −18.71816470150515131610888731909, −17.82347816457511707689638697177, −17.46441703004282845162360960588, −16.44891361038071800053760456457, −15.66970709792428017682473559104, −15.012934785821065386726330163365, −14.3148242685906112155223119028, −13.27123593448137745911112498002, −12.997039044080915157650448554313, −11.69172190623010006433477941758, −10.40747954929272064689169853081, −9.894981799111742741549989637695, −9.23860055868571776921867880932, −8.178495256060335544335955881180, −7.261527595516120715313233477256, −6.55039611905069989297020124527, −5.79273116512404806602633675642, −5.15913020697661052499691130233, −3.72546412287796429225920769568, −2.93966333394778742463201417755, −1.60651700724967570070359939154, 0.148249467288252605412384169896, 1.392625253984030441271818566219, 2.364050522795239969358651616037, 3.255865366122689552106893941981, 4.38502976887032106303921592649, 5.024278479979590332767550100009, 6.17005106236706716779069309079, 7.17312022102879347309057122012, 8.340672852412207577860836131224, 9.20434373034738151983121741187, 9.77507148874031013157059112685, 10.28291404507352797091288301983, 11.62187259577316644759540575922, 12.1042135311684186217041079069, 13.01557925120951388722973072346, 13.68982605081845197090339202370, 14.1385637105856339023484745231, 15.615366769789464609597551222132, 16.639853311335042856996838620144, 16.92725181464245026989461086310, 17.991740784221349893767283529535, 18.66970574874232245798995467526, 19.48307326664218197889853143258, 20.337638318645421064085308303211, 20.62467086401556876225969644962

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