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Properties

Label 1-33e2-1089.455-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.0875 - 0.996i$
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  + (−0.997 + 0.0760i)2-s + (0.988 − 0.151i)4-s + (−0.797 − 0.603i)5-s + (0.861 − 0.508i)7-s + (−0.974 + 0.226i)8-s + (0.841 + 0.540i)10-s + (0.161 − 0.986i)13-s + (−0.820 + 0.572i)14-s + (0.953 − 0.299i)16-s + (−0.897 − 0.441i)17-s + (−0.466 + 0.884i)19-s + (−0.879 − 0.475i)20-s + (0.995 − 0.0950i)23-s + (0.272 + 0.962i)25-s + (−0.0855 + 0.996i)26-s + ⋯
L(s)  = 1  + (−0.997 + 0.0760i)2-s + (0.988 − 0.151i)4-s + (−0.797 − 0.603i)5-s + (0.861 − 0.508i)7-s + (−0.974 + 0.226i)8-s + (0.841 + 0.540i)10-s + (0.161 − 0.986i)13-s + (−0.820 + 0.572i)14-s + (0.953 − 0.299i)16-s + (−0.897 − 0.441i)17-s + (−0.466 + 0.884i)19-s + (−0.879 − 0.475i)20-s + (0.995 − 0.0950i)23-s + (0.272 + 0.962i)25-s + (−0.0855 + 0.996i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0875 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0875 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.0875 - 0.996i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (455, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.0875 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7924421267 - 0.8651745688i\)
\(L(\frac12)\) \(\approx\) \(0.7924421267 - 0.8651745688i\)
\(L(1)\) \(\approx\) \(0.6861156103 - 0.1823901536i\)
\(L(1)\) \(\approx\) \(0.6861156103 - 0.1823901536i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.997 + 0.0760i)T \)
5 \( 1 + (-0.797 - 0.603i)T \)
7 \( 1 + (0.861 - 0.508i)T \)
13 \( 1 + (0.161 - 0.986i)T \)
17 \( 1 + (-0.897 - 0.441i)T \)
19 \( 1 + (-0.466 + 0.884i)T \)
23 \( 1 + (0.995 - 0.0950i)T \)
29 \( 1 + (0.969 - 0.244i)T \)
31 \( 1 + (0.964 - 0.263i)T \)
37 \( 1 + (-0.998 + 0.0570i)T \)
41 \( 1 + (0.179 + 0.983i)T \)
43 \( 1 + (-0.327 - 0.945i)T \)
47 \( 1 + (0.991 - 0.132i)T \)
53 \( 1 + (0.736 - 0.676i)T \)
59 \( 1 + (0.761 + 0.647i)T \)
61 \( 1 + (-0.432 - 0.901i)T \)
67 \( 1 + (0.723 + 0.690i)T \)
71 \( 1 + (0.985 + 0.170i)T \)
73 \( 1 + (0.198 + 0.980i)T \)
79 \( 1 + (0.483 + 0.875i)T \)
83 \( 1 + (0.398 - 0.917i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.797 - 0.603i)T \)
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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

−21.3657895343778971053865776596, −20.59491223329195087187057794511, −19.45719609328224255136397960936, −19.27199158380487475284286843297, −18.347113200409482257047576158130, −17.67659667517870825958644972841, −17.00187113487718613656981168732, −15.86801664868245200599041307685, −15.38826051201396955354007596761, −14.71993814873599821836542495328, −13.712950071536308381169098580533, −12.32758673742964484095124568791, −11.759359690237875153955224280035, −10.9472626041439422073039165538, −10.59933594225658730082876493581, −9.15092429695669374688635258587, −8.67713652048015074907974159745, −7.89176126897855509906420775899, −6.892257830484471687192634459125, −6.45964522566102626423090220273, −4.98478948155148915611299304381, −4.01445331732943215991707353206, −2.79156244570670794827363388752, −2.06336472106582564705448026737, −0.86006597947936220665605842794, 0.47926300569830909889252597490, 1.129636886799639995756250709466, 2.34051570100126203202229751756, 3.52835855406911866202610126283, 4.601530530275656489779748976472, 5.47803941409748564734534709349, 6.73373492088101856011345055362, 7.48200422822606876523047080685, 8.43939703606193356110593889893, 8.535626306390035348079096079649, 9.9059015290141534714001506991, 10.670047102501435459930076792628, 11.372059175905822038566002177092, 12.09449987386588772472407326432, 12.99856154366900976529728077700, 14.10744107762907401432878531715, 15.22837854109294211998913979985, 15.52167016105560707190351862541, 16.55854535641227363114964644262, 17.2056365814546623274839018818, 17.81805895466879334235370837359, 18.732420735438514069287649158687, 19.48362357243105077267693827148, 20.285979285502678540000148564415, 20.65614992785100921658852743167

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