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Properties

Label 1-33e2-1089.29-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.906 - 0.423i$
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  + (−0.432 + 0.901i)2-s + (−0.625 − 0.780i)4-s + (−0.123 + 0.992i)5-s + (−0.00951 − 0.999i)7-s + (0.974 − 0.226i)8-s + (−0.841 − 0.540i)10-s + (0.935 − 0.353i)13-s + (0.905 + 0.424i)14-s + (−0.217 + 0.976i)16-s + (0.897 + 0.441i)17-s + (0.466 − 0.884i)19-s + (0.851 − 0.524i)20-s + (−0.580 − 0.814i)23-s + (−0.969 − 0.244i)25-s + (−0.0855 + 0.996i)26-s + ⋯
L(s)  = 1  + (−0.432 + 0.901i)2-s + (−0.625 − 0.780i)4-s + (−0.123 + 0.992i)5-s + (−0.00951 − 0.999i)7-s + (0.974 − 0.226i)8-s + (−0.841 − 0.540i)10-s + (0.935 − 0.353i)13-s + (0.905 + 0.424i)14-s + (−0.217 + 0.976i)16-s + (0.897 + 0.441i)17-s + (0.466 − 0.884i)19-s + (0.851 − 0.524i)20-s + (−0.580 − 0.814i)23-s + (−0.969 − 0.244i)25-s + (−0.0855 + 0.996i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.906 - 0.423i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.906 - 0.423i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8644592825 - 0.1918422481i\)
\(L(\frac12)\) \(\approx\) \(0.8644592825 - 0.1918422481i\)
\(L(1)\) \(\approx\) \(0.7681375784 + 0.1851741195i\)
\(L(1)\) \(\approx\) \(0.7681375784 + 0.1851741195i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.432 + 0.901i)T \)
5 \( 1 + (-0.123 + 0.992i)T \)
7 \( 1 + (-0.00951 - 0.999i)T \)
13 \( 1 + (0.935 - 0.353i)T \)
17 \( 1 + (0.897 + 0.441i)T \)
19 \( 1 + (0.466 - 0.884i)T \)
23 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (0.272 - 0.962i)T \)
31 \( 1 + (-0.710 - 0.703i)T \)
37 \( 1 + (-0.998 + 0.0570i)T \)
41 \( 1 + (-0.761 + 0.647i)T \)
43 \( 1 + (-0.981 - 0.189i)T \)
47 \( 1 + (-0.380 + 0.924i)T \)
53 \( 1 + (0.736 - 0.676i)T \)
59 \( 1 + (0.179 - 0.983i)T \)
61 \( 1 + (-0.997 - 0.0760i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (0.985 + 0.170i)T \)
73 \( 1 + (-0.198 - 0.980i)T \)
79 \( 1 + (0.999 + 0.0190i)T \)
83 \( 1 + (-0.595 - 0.803i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.123 + 0.992i)T \)
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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

−21.3304136029214202049420368406, −20.71117414296218299226360519063, −20.00388815265290119538489864305, −19.20339962733012247793930060739, −18.35968856354325188298554153992, −17.983280134011820522888811994220, −16.719004192754930566980084409216, −16.31525947376039311427828663969, −15.4559526874818300049784514376, −14.13223172752634238704465534222, −13.492294286105361746775068416373, −12.41528200074475672332120423564, −12.10055048471960019546470446590, −11.37456955155876381871131569304, −10.23389747571227927985666116482, −9.47352704968372717785281645351, −8.68478854393711847547211268829, −8.26282498262804989349866602370, −7.15448579452139785309670951307, −5.61832064621235866357482477190, −5.14300198324012564792685964077, −3.85351157048086497337785270426, −3.22533510605685454901355993858, −1.83257744864823143425355814088, −1.26146342742699130644476089728, 0.47004562838259744711360291998, 1.736857356146490904146804467495, 3.278905511401621335452561948, 4.02954965970260494086838821669, 5.14784216485655191224547153128, 6.265702508163701914193382567663, 6.71073674427344605315093927302, 7.74270538300985889398139631556, 8.1414741136106544650364501174, 9.41894694358812576943625182029, 10.26399604442777355384662401805, 10.71742861861987326327558282725, 11.65597544558464308353243344109, 13.11149487196210838663823975777, 13.751897480939650284030466492432, 14.41858408472202651613099291031, 15.19421491871719752418344135209, 15.92158044953561315421843915013, 16.736851296896608036112680406900, 17.483468283560052511859827926624, 18.23853648950004387422737556369, 18.81768454079737107527113704104, 19.675926323086767534838771499446, 20.391665929198949788810724437055, 21.50438517963718783251695453212

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