Properties

Label 1-33e2-1089.383-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.321 - 0.946i$
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  + (−0.988 − 0.151i)2-s + (0.953 + 0.299i)4-s + (−0.272 + 0.962i)5-s + (0.483 + 0.875i)7-s + (−0.897 − 0.441i)8-s + (0.415 − 0.909i)10-s + (−0.948 + 0.318i)13-s + (−0.345 − 0.938i)14-s + (0.820 + 0.572i)16-s + (−0.610 + 0.791i)17-s + (−0.564 + 0.825i)19-s + (−0.548 + 0.836i)20-s + (−0.981 − 0.189i)23-s + (−0.851 − 0.524i)25-s + (0.985 − 0.170i)26-s + ⋯
L(s)  = 1  + (−0.988 − 0.151i)2-s + (0.953 + 0.299i)4-s + (−0.272 + 0.962i)5-s + (0.483 + 0.875i)7-s + (−0.897 − 0.441i)8-s + (0.415 − 0.909i)10-s + (−0.948 + 0.318i)13-s + (−0.345 − 0.938i)14-s + (0.820 + 0.572i)16-s + (−0.610 + 0.791i)17-s + (−0.564 + 0.825i)19-s + (−0.548 + 0.836i)20-s + (−0.981 − 0.189i)23-s + (−0.851 − 0.524i)25-s + (0.985 − 0.170i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.321 - 0.946i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.321 - 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07731604271 - 0.05537646907i\)
\(L(\frac12)\) \(\approx\) \(0.07731604271 - 0.05537646907i\)
\(L(1)\) \(\approx\) \(0.5191637167 + 0.1779431909i\)
\(L(1)\) \(\approx\) \(0.5191637167 + 0.1779431909i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.988 - 0.151i)T \)
5 \( 1 + (-0.272 + 0.962i)T \)
7 \( 1 + (0.483 + 0.875i)T \)
13 \( 1 + (-0.948 + 0.318i)T \)
17 \( 1 + (-0.610 + 0.791i)T \)
19 \( 1 + (-0.564 + 0.825i)T \)
23 \( 1 + (-0.981 - 0.189i)T \)
29 \( 1 + (-0.879 - 0.475i)T \)
31 \( 1 + (0.861 + 0.508i)T \)
37 \( 1 + (0.993 + 0.113i)T \)
41 \( 1 + (0.935 + 0.353i)T \)
43 \( 1 + (-0.786 - 0.618i)T \)
47 \( 1 + (-0.964 - 0.263i)T \)
53 \( 1 + (-0.0855 - 0.996i)T \)
59 \( 1 + (-0.161 + 0.986i)T \)
61 \( 1 + (-0.625 - 0.780i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (-0.941 + 0.336i)T \)
73 \( 1 + (-0.921 - 0.389i)T \)
79 \( 1 + (-0.532 - 0.846i)T \)
83 \( 1 + (0.683 - 0.730i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (0.272 + 0.962i)T \)
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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

−21.06310099064894621448316746976, −20.29854432017851775824082821797, −19.913217596036122598633023529, −19.24524347669537694251142443906, −18.022013043946886916957521822921, −17.51303351995279649350673589129, −16.811022137198528416624890369426, −16.18841277272941831073168655497, −15.348290426875133540914961886088, −14.549543855053791992209733166615, −13.49026804517496529957730509610, −12.626606234481985916339736674623, −11.61948481423313491220474550093, −11.10935134609246709730966536466, −10.017573424119437146165128220304, −9.40186010542440434976213294467, −8.50513462830135618730969823385, −7.71426176664632728236514908646, −7.18930545752644805982838332857, −6.03759129249247118923495830865, −4.9304797770663334071553210945, −4.24081271725956193046073694495, −2.7649451131672612003056465621, −1.7225462251233771632982641745, −0.67773477727484187629451351005, 0.03753283614932972076268118661, 1.84349649659406651547222671584, 2.297441215914067181203386513, 3.368658527721140502558143146770, 4.48702465405780396289129840053, 5.98863658839177608294448174930, 6.46488462699960352392742892193, 7.63079910849048884126426441078, 8.11750326141363677294141397985, 9.05764184813013477954665035848, 10.005200542902117945520561668936, 10.59641166443839386582134362536, 11.61161287593812698569942520498, 11.95049620331229934565624287807, 12.9878386967617018144851390534, 14.488726926778266616021880551809, 14.86997947243673865483110308534, 15.61793289460229164040501464481, 16.53439231448087790065681138051, 17.44463613184633710156671127752, 18.05876570759322846749732486276, 18.77059637141319264419762081947, 19.343241776617546470256159978, 20.040791191668420505043643687957, 21.16196123657198318905910794822

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