Properties

Label 1-33e2-1089.1055-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.122 - 0.992i$
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  + (0.723 + 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.981 + 0.189i)5-s + (−0.580 + 0.814i)7-s + (−0.654 + 0.755i)8-s + (−0.841 − 0.540i)10-s + (−0.0475 − 0.998i)13-s + (−0.981 + 0.189i)14-s + (−0.995 + 0.0950i)16-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.235 − 0.971i)20-s + (−0.580 − 0.814i)23-s + (0.928 − 0.371i)25-s + (0.654 − 0.755i)26-s + ⋯
L(s)  = 1  + (0.723 + 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.981 + 0.189i)5-s + (−0.580 + 0.814i)7-s + (−0.654 + 0.755i)8-s + (−0.841 − 0.540i)10-s + (−0.0475 − 0.998i)13-s + (−0.981 + 0.189i)14-s + (−0.995 + 0.0950i)16-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.235 − 0.971i)20-s + (−0.580 − 0.814i)23-s + (0.928 − 0.371i)25-s + (0.654 − 0.755i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.122 - 0.992i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.122 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1758722800 + 0.1988741385i\)
\(L(\frac12)\) \(\approx\) \(-0.1758722800 + 0.1988741385i\)
\(L(1)\) \(\approx\) \(0.7357283863 + 0.5503662447i\)
\(L(1)\) \(\approx\) \(0.7357283863 + 0.5503662447i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.723 + 0.690i)T \)
5 \( 1 + (-0.981 + 0.189i)T \)
7 \( 1 + (-0.580 + 0.814i)T \)
13 \( 1 + (-0.0475 - 0.998i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (-0.786 + 0.618i)T \)
31 \( 1 + (-0.888 + 0.458i)T \)
37 \( 1 + (0.841 + 0.540i)T \)
41 \( 1 + (0.235 - 0.971i)T \)
43 \( 1 + (-0.981 - 0.189i)T \)
47 \( 1 + (-0.235 - 0.971i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (-0.723 + 0.690i)T \)
61 \( 1 + (-0.235 - 0.971i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (-0.415 + 0.909i)T \)
79 \( 1 + (0.327 - 0.945i)T \)
83 \( 1 + (0.580 - 0.814i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.981 + 0.189i)T \)
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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

−20.72990533535825311805870932860, −20.045624606485987282537337933840, −19.600587331486899506524586926782, −18.86612025467895486426483534722, −18.01272166963200336200230132796, −16.69184501430118567353467575469, −16.10502647943446826849769454387, −15.3447690642532740835894531224, −14.46590836902583847415680369418, −13.58471146950943250414439862755, −13.0794531855633483058893428230, −12.077637629561973148141482026675, −11.361835569344885455470690903760, −10.92846397348374048498130422580, −9.545923749208864746211994052898, −9.30048920561409383188918641714, −7.71501873585959444544376015623, −7.04129832991820617788342011689, −6.13777811112291330268911547836, −4.88712159549541754435952512274, −4.23697577492433207685673586348, −3.52889573832901280672654308910, −2.58738935012910256810723227900, −1.27176929169272071846090844491, −0.08683367695556053067852886112, 2.10119669923722142477570042073, 3.3186451499946911991247582647, 3.67627293836824878182545733424, 4.87720216490247533868024410309, 5.76387483721851693663706069490, 6.464804870684530030650345284917, 7.45475563483261640507626799727, 8.222129175681129806686578612580, 8.79369070139897638128456586819, 10.13212955218727194582924813426, 11.09101377327418926597732608088, 12.11190399443829482450707657933, 12.5424855739986760807610455913, 13.212686359646901648289915151768, 14.58947024124036322824101345602, 14.88205320772991720668795426041, 15.71948963830489480311688837694, 16.28418159356396734270798410441, 17.06434077944782095362882312444, 18.254035833049768572820372749294, 18.69014651647889044344475371536, 19.90738461820879814390895872206, 20.36238886057863313979069890155, 21.565564007718910384197662309257, 22.186001123208075802617693190063

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