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Properties

Label 1-33e2-1089.223-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.981 + 0.190i$
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  + (0.879 − 0.475i)2-s + (0.548 − 0.836i)4-s + (0.483 + 0.875i)5-s + (−0.948 + 0.318i)7-s + (0.0855 − 0.996i)8-s + (0.841 + 0.540i)10-s + (0.964 + 0.263i)13-s + (−0.683 + 0.730i)14-s + (−0.398 − 0.917i)16-s + (−0.985 + 0.170i)17-s + (0.696 + 0.717i)19-s + (0.997 + 0.0760i)20-s + (0.580 + 0.814i)23-s + (−0.532 + 0.846i)25-s + (0.974 − 0.226i)26-s + ⋯
L(s)  = 1  + (0.879 − 0.475i)2-s + (0.548 − 0.836i)4-s + (0.483 + 0.875i)5-s + (−0.948 + 0.318i)7-s + (0.0855 − 0.996i)8-s + (0.841 + 0.540i)10-s + (0.964 + 0.263i)13-s + (−0.683 + 0.730i)14-s + (−0.398 − 0.917i)16-s + (−0.985 + 0.170i)17-s + (0.696 + 0.717i)19-s + (0.997 + 0.0760i)20-s + (0.580 + 0.814i)23-s + (−0.532 + 0.846i)25-s + (0.974 − 0.226i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.981 + 0.190i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.981 + 0.190i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.599988997 + 0.2497827203i\)
\(L(\frac12)\) \(\approx\) \(2.599988997 + 0.2497827203i\)
\(L(1)\) \(\approx\) \(1.778932075 - 0.1212293555i\)
\(L(1)\) \(\approx\) \(1.778932075 - 0.1212293555i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.879 - 0.475i)T \)
5 \( 1 + (0.483 + 0.875i)T \)
7 \( 1 + (-0.948 + 0.318i)T \)
13 \( 1 + (0.964 + 0.263i)T \)
17 \( 1 + (-0.985 + 0.170i)T \)
19 \( 1 + (0.696 + 0.717i)T \)
23 \( 1 + (0.580 + 0.814i)T \)
29 \( 1 + (0.999 - 0.0380i)T \)
31 \( 1 + (0.161 + 0.986i)T \)
37 \( 1 + (-0.362 - 0.931i)T \)
41 \( 1 + (0.380 + 0.924i)T \)
43 \( 1 + (0.981 + 0.189i)T \)
47 \( 1 + (-0.761 - 0.647i)T \)
53 \( 1 + (0.993 - 0.113i)T \)
59 \( 1 + (-0.991 + 0.132i)T \)
61 \( 1 + (-0.851 + 0.524i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (0.897 - 0.441i)T \)
73 \( 1 + (-0.870 + 0.491i)T \)
79 \( 1 + (0.797 + 0.603i)T \)
83 \( 1 + (0.953 + 0.299i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (0.483 - 0.875i)T \)
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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

−21.51259470333371845218573296526, −20.56277967712313577358583449487, −20.28741237332984674825862909573, −19.28625962963802838642938334119, −18.03975948824210791710128188279, −17.30609365335938049512624192029, −16.57180225246435030879417413712, −15.83151839162376727740046013853, −15.439140882008040207518545708586, −14.05987959521892260856827701589, −13.488599369264174004081721478862, −12.994551339031978421749968091, −12.25391843650236123595222142137, −11.259136722901599907035918509469, −10.33746990026731139225025937392, −9.16257558903783770408636562651, −8.61106142392465177947906057973, −7.49612079993088335837448853086, −6.50240932665113029052714313225, −6.01911100960148618974872691800, −4.93869805981675222082289900771, −4.27039726091004784063911617298, −3.219038179139362187289256138867, −2.33056170371556968501271437976, −0.85674494693758390194757633295, 1.31464368906432824373635686979, 2.370301576580231465396693089432, 3.21256578154009860904492599819, 3.81869015778815516874225704832, 5.10584338046664923498414369923, 6.08424609320194766936368218320, 6.477283732372481162368555602946, 7.37509009893075385202170726931, 8.91585371268033576774434959129, 9.67994589474219055412866423264, 10.50850479987092554801371073238, 11.1379600786967760441285826330, 12.0285946427589869401689180290, 12.90777381417582536404799715145, 13.65263895582958493676485200145, 14.12771926218791613470975256741, 15.205525245641685815626631279064, 15.735784604054340187710847746963, 16.52288447607147894732667515124, 17.9273576588902411301578581622, 18.40071245712371823315048381111, 19.43410496648447906584695987654, 19.733137455523905124081299885339, 21.04013036780305880586701288737, 21.45884674828371660463798087107

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