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Properties

Label 1-33e2-1089.1031-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.999 + 0.0161i$
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  + (−0.999 + 0.0190i)2-s + (0.999 − 0.0380i)4-s + (−0.161 + 0.986i)5-s + (0.991 − 0.132i)7-s + (−0.998 + 0.0570i)8-s + (0.142 − 0.989i)10-s + (−0.345 − 0.938i)13-s + (−0.988 + 0.151i)14-s + (0.997 − 0.0760i)16-s + (0.993 + 0.113i)17-s + (0.870 + 0.491i)19-s + (−0.123 + 0.992i)20-s + (−0.723 − 0.690i)23-s + (−0.948 − 0.318i)25-s + (0.362 + 0.931i)26-s + ⋯
L(s)  = 1  + (−0.999 + 0.0190i)2-s + (0.999 − 0.0380i)4-s + (−0.161 + 0.986i)5-s + (0.991 − 0.132i)7-s + (−0.998 + 0.0570i)8-s + (0.142 − 0.989i)10-s + (−0.345 − 0.938i)13-s + (−0.988 + 0.151i)14-s + (0.997 − 0.0760i)16-s + (0.993 + 0.113i)17-s + (0.870 + 0.491i)19-s + (−0.123 + 0.992i)20-s + (−0.723 − 0.690i)23-s + (−0.948 − 0.318i)25-s + (0.362 + 0.931i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.0161i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.999 + 0.0161i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.082495327 + 0.008744107740i\)
\(L(\frac12)\) \(\approx\) \(1.082495327 + 0.008744107740i\)
\(L(1)\) \(\approx\) \(0.8102297500 + 0.05515886391i\)
\(L(1)\) \(\approx\) \(0.8102297500 + 0.05515886391i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.999 + 0.0190i)T \)
5 \( 1 + (-0.161 + 0.986i)T \)
7 \( 1 + (0.991 - 0.132i)T \)
13 \( 1 + (-0.345 - 0.938i)T \)
17 \( 1 + (0.993 + 0.113i)T \)
19 \( 1 + (0.870 + 0.491i)T \)
23 \( 1 + (-0.723 - 0.690i)T \)
29 \( 1 + (0.749 + 0.662i)T \)
31 \( 1 + (-0.0665 - 0.997i)T \)
37 \( 1 + (0.696 - 0.717i)T \)
41 \( 1 + (-0.905 + 0.424i)T \)
43 \( 1 + (0.888 - 0.458i)T \)
47 \( 1 + (0.683 - 0.730i)T \)
53 \( 1 + (0.564 - 0.825i)T \)
59 \( 1 + (-0.820 + 0.572i)T \)
61 \( 1 + (-0.483 - 0.875i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (0.736 - 0.676i)T \)
73 \( 1 + (-0.941 - 0.336i)T \)
79 \( 1 + (-0.964 - 0.263i)T \)
83 \( 1 + (0.879 + 0.475i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 + (0.161 + 0.986i)T \)
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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

−21.342111786457018391211487861972, −20.45898394926453928792411105832, −19.93461010189498609939318693023, −19.06755945971656968994858010193, −18.307517792310408262131938801259, −17.433237170111607770701159395216, −16.98248943631770258855669689279, −16.0594505399087962123151650632, −15.54824880578321464390751136757, −14.42291300959482128383182971539, −13.709242343904275107616649100260, −12.27626334515116500695973460364, −11.90514054367862532354390095704, −11.208444303962114985459524207835, −10.04769067051667852503640247041, −9.360577187493495832564652652, −8.60626908445134451086741929766, −7.846696972202203803809866055630, −7.23962184673705698740424701422, −5.94611762552148412804244694276, −5.111757141970033123533895812944, −4.16348186490513682400948992815, −2.819723389636828510177698423121, −1.66281120171216078909356934959, −1.01841420848693633759703089091, 0.796931152063888854148039764445, 1.996139327632651116599182725653, 2.874774234811983246471950441064, 3.822820704103940562503491527532, 5.31017161031464379691530024317, 6.08526805792568450439748911216, 7.20591997354652835256506209988, 7.79120252128364873976713903865, 8.32807945239226145810937675098, 9.63156617872582624719915333063, 10.33020255841959269626011620864, 10.86423433021473446783745460529, 11.79302437470334003646562922902, 12.35543984818150616179845561848, 13.88733326969926358701381013104, 14.6288709821677686656415810149, 15.14228018534145255387457921935, 16.08797226083121176628918935544, 16.9348431403274036179725948128, 17.7772580830086218328966475025, 18.30547903916821923636151324278, 18.841833645519078717769998935006, 19.93422009215377660265080959547, 20.391578086171234864375293634144, 21.31908287471078561942774888303

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