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Properties

Label 1-33e2-1089.131-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.777 - 0.629i$
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  + (−0.995 − 0.0950i)2-s + (0.981 + 0.189i)4-s + (−0.723 − 0.690i)5-s + (0.786 + 0.618i)7-s + (−0.959 − 0.281i)8-s + (0.654 + 0.755i)10-s + (−0.981 − 0.189i)13-s + (−0.723 − 0.690i)14-s + (0.928 + 0.371i)16-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (−0.580 − 0.814i)20-s + (0.786 − 0.618i)23-s + (0.0475 + 0.998i)25-s + (0.959 + 0.281i)26-s + ⋯
L(s)  = 1  + (−0.995 − 0.0950i)2-s + (0.981 + 0.189i)4-s + (−0.723 − 0.690i)5-s + (0.786 + 0.618i)7-s + (−0.959 − 0.281i)8-s + (0.654 + 0.755i)10-s + (−0.981 − 0.189i)13-s + (−0.723 − 0.690i)14-s + (0.928 + 0.371i)16-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (−0.580 − 0.814i)20-s + (0.786 − 0.618i)23-s + (0.0475 + 0.998i)25-s + (0.959 + 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.777 - 0.629i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.777 - 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1363881311 - 0.3850599419i\)
\(L(\frac12)\) \(\approx\) \(0.1363881311 - 0.3850599419i\)
\(L(1)\) \(\approx\) \(0.5518512007 - 0.1259959186i\)
\(L(1)\) \(\approx\) \(0.5518512007 - 0.1259959186i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.995 - 0.0950i)T \)
5 \( 1 + (-0.723 - 0.690i)T \)
7 \( 1 + (0.786 + 0.618i)T \)
13 \( 1 + (-0.981 - 0.189i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (-0.888 + 0.458i)T \)
31 \( 1 + (-0.327 + 0.945i)T \)
37 \( 1 + (-0.654 - 0.755i)T \)
41 \( 1 + (0.580 - 0.814i)T \)
43 \( 1 + (-0.723 + 0.690i)T \)
47 \( 1 + (-0.580 - 0.814i)T \)
53 \( 1 + (0.142 - 0.989i)T \)
59 \( 1 + (0.995 - 0.0950i)T \)
61 \( 1 + (-0.580 - 0.814i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (-0.235 + 0.971i)T \)
83 \( 1 + (-0.786 - 0.618i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.723 - 0.690i)T \)
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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

−21.49201080770346378798314508794, −20.818364797074626556607587919567, −20.00182801519155065789952661346, −19.17100730304469398387140707729, −18.83739897766629347820000483937, −17.84750352439294102677052857646, −16.988907527983450369177791424837, −16.6836524050921570189877346831, −15.391866906553796066451251816615, −14.836005318521639608064049865087, −14.34679237751577441097598165987, −12.93774027022155593403750550355, −11.83109195760827334308164035678, −11.38236920084285689990355816851, −10.47573160212985945069115026132, −9.96515264069683542007697240546, −8.81282025101300455059769436167, −7.79154630616075887905041665691, −7.55979338003659394877614059062, −6.638994660345485029042756881428, −5.58490362158018618207623771081, −4.343329044131447458798205727737, −3.37345457009660954429525738080, −2.27874436159878986972455138775, −1.26329684696800375862494141197, 0.248438509433575702113388599188, 1.49254705607837163271142928657, 2.45788762328043358902486545458, 3.51727455948998680119153204588, 4.86177093518929560655032796876, 5.44870559473258747566688954556, 6.90798274983667623792460311107, 7.53765424817647260847746357463, 8.43629899641589441537997075416, 8.90541413917621791440733316565, 9.8076709820171251680065508750, 10.88992379183680066591833594146, 11.49124870301483387448380270278, 12.36082585165705129457183547018, 12.762844297874733769563173293383, 14.47063733442875082437336446569, 14.99790577563964742715736234758, 15.81880853505290536761199728729, 16.62665032843521320079285614727, 17.22095640465036739845149483506, 18.05913356678506144132028215446, 18.85819834695298569363873828732, 19.51490157708062503563419027485, 20.20404707513948496531962587248, 21.04862342041163836542413308174

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