Properties

Label 1-33e2-1089.194-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.945 + 0.326i$
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  + (0.483 + 0.875i)2-s + (−0.532 + 0.846i)4-s + (0.935 + 0.353i)5-s + (−0.380 − 0.924i)7-s + (−0.998 − 0.0570i)8-s + (0.142 + 0.989i)10-s + (−0.640 − 0.768i)13-s + (0.625 − 0.780i)14-s + (−0.432 − 0.901i)16-s + (0.993 − 0.113i)17-s + (0.870 − 0.491i)19-s + (−0.797 + 0.603i)20-s + (−0.235 − 0.971i)23-s + (0.749 + 0.662i)25-s + (0.362 − 0.931i)26-s + ⋯
L(s)  = 1  + (0.483 + 0.875i)2-s + (−0.532 + 0.846i)4-s + (0.935 + 0.353i)5-s + (−0.380 − 0.924i)7-s + (−0.998 − 0.0570i)8-s + (0.142 + 0.989i)10-s + (−0.640 − 0.768i)13-s + (0.625 − 0.780i)14-s + (−0.432 − 0.901i)16-s + (0.993 − 0.113i)17-s + (0.870 − 0.491i)19-s + (−0.797 + 0.603i)20-s + (−0.235 − 0.971i)23-s + (0.749 + 0.662i)25-s + (0.362 − 0.931i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.945 + 0.326i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.945 + 0.326i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.884945558 + 0.3166896050i\)
\(L(\frac12)\) \(\approx\) \(1.884945558 + 0.3166896050i\)
\(L(1)\) \(\approx\) \(1.317366266 + 0.4376584125i\)
\(L(1)\) \(\approx\) \(1.317366266 + 0.4376584125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.483 + 0.875i)T \)
5 \( 1 + (0.935 + 0.353i)T \)
7 \( 1 + (-0.380 - 0.924i)T \)
13 \( 1 + (-0.640 - 0.768i)T \)
17 \( 1 + (0.993 - 0.113i)T \)
19 \( 1 + (0.870 - 0.491i)T \)
23 \( 1 + (-0.235 - 0.971i)T \)
29 \( 1 + (-0.948 - 0.318i)T \)
31 \( 1 + (-0.830 - 0.556i)T \)
37 \( 1 + (0.696 + 0.717i)T \)
41 \( 1 + (0.820 - 0.572i)T \)
43 \( 1 + (-0.0475 - 0.998i)T \)
47 \( 1 + (0.290 - 0.956i)T \)
53 \( 1 + (0.564 + 0.825i)T \)
59 \( 1 + (0.905 - 0.424i)T \)
61 \( 1 + (0.999 - 0.0190i)T \)
67 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (0.736 + 0.676i)T \)
73 \( 1 + (-0.941 + 0.336i)T \)
79 \( 1 + (0.710 + 0.703i)T \)
83 \( 1 + (-0.851 - 0.524i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (-0.935 + 0.353i)T \)
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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

−21.32637718622977496797439456819, −20.927144485198607838852153946540, −19.83252980104858559275158084917, −19.235265459579576895248340186418, −18.31295415894032181963796156845, −17.8783598773180066620111648571, −16.6629341300885125989339698523, −15.99589393650529777584859042809, −14.64105481661417855771976630947, −14.411743630615310630675599285741, −13.33966620792046369745634265654, −12.677325162134430638326957208967, −12.031694269347111166747612150929, −11.260568432305100134754894426314, −10.09985284238851915312281004969, −9.43542826876372877337192555590, −9.12552527234349055375001038726, −7.740369221597713002620139510775, −6.406068148485015240785974472141, −5.55411922598370513196120220259, −5.1882952334835199227199081661, −3.89696307916516939333211804986, −2.93522155827004727042033402848, −2.0577415153659730096853779003, −1.268402317527560735387318921120, 0.71792907866883491090157376811, 2.43777037626499236536933735914, 3.29305343568110446315451920041, 4.23278783786026393033938013931, 5.40807407901112265348433916581, 5.820661901299258775129089497979, 7.08008999761424199193756993303, 7.31624168572123118223211664958, 8.459099542825917664635300612229, 9.6438661878743686988048138732, 10.02159349897310782017049137701, 11.1532406896311354463531661901, 12.34643287812197475893019955232, 13.08093872229261242330471492620, 13.70315625929473613116644083448, 14.4377992615985127975126535420, 15.02960751215354702311530387551, 16.13181742095267554811160032591, 16.84578542218391940367411638665, 17.32930243737946047864709824200, 18.19256778292649449444548180476, 18.88364794505861151559922816635, 20.28289447517120840258479544657, 20.66415149802123638578582983677, 21.83160982678045318765671611083

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