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Properties

Label 1-33e2-1089.292-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.920 - 0.391i$
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  + (−0.380 − 0.924i)2-s + (−0.710 + 0.703i)4-s + (0.820 − 0.572i)5-s + (0.398 − 0.917i)7-s + (0.921 + 0.389i)8-s + (−0.841 − 0.540i)10-s + (−0.988 + 0.151i)13-s + (−0.999 − 0.0190i)14-s + (0.00951 − 0.999i)16-s + (−0.696 + 0.717i)17-s + (−0.897 + 0.441i)19-s + (−0.179 + 0.983i)20-s + (−0.995 + 0.0950i)23-s + (0.345 − 0.938i)25-s + (0.516 + 0.856i)26-s + ⋯
L(s)  = 1  + (−0.380 − 0.924i)2-s + (−0.710 + 0.703i)4-s + (0.820 − 0.572i)5-s + (0.398 − 0.917i)7-s + (0.921 + 0.389i)8-s + (−0.841 − 0.540i)10-s + (−0.988 + 0.151i)13-s + (−0.999 − 0.0190i)14-s + (0.00951 − 0.999i)16-s + (−0.696 + 0.717i)17-s + (−0.897 + 0.441i)19-s + (−0.179 + 0.983i)20-s + (−0.995 + 0.0950i)23-s + (0.345 − 0.938i)25-s + (0.516 + 0.856i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.920 - 0.391i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (292, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.920 - 0.391i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.312589215 - 0.2673450583i\)
\(L(\frac12)\) \(\approx\) \(1.312589215 - 0.2673450583i\)
\(L(1)\) \(\approx\) \(0.7918653071 - 0.3955935936i\)
\(L(1)\) \(\approx\) \(0.7918653071 - 0.3955935936i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.380 - 0.924i)T \)
5 \( 1 + (0.820 - 0.572i)T \)
7 \( 1 + (0.398 - 0.917i)T \)
13 \( 1 + (-0.988 + 0.151i)T \)
17 \( 1 + (-0.696 + 0.717i)T \)
19 \( 1 + (-0.897 + 0.441i)T \)
23 \( 1 + (-0.995 + 0.0950i)T \)
29 \( 1 + (-0.640 + 0.768i)T \)
31 \( 1 + (0.548 + 0.836i)T \)
37 \( 1 + (0.774 - 0.633i)T \)
41 \( 1 + (0.432 - 0.901i)T \)
43 \( 1 + (0.327 + 0.945i)T \)
47 \( 1 + (0.879 + 0.475i)T \)
53 \( 1 + (-0.870 - 0.491i)T \)
59 \( 1 + (0.997 + 0.0760i)T \)
61 \( 1 + (0.991 - 0.132i)T \)
67 \( 1 + (0.723 + 0.690i)T \)
71 \( 1 + (-0.466 + 0.884i)T \)
73 \( 1 + (0.736 + 0.676i)T \)
79 \( 1 + (0.683 - 0.730i)T \)
83 \( 1 + (-0.749 - 0.662i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (0.820 + 0.572i)T \)
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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

−21.5840867221818694802859591899, −20.50087090376059717410512169713, −19.44710137330848908109301605278, −18.677882151349588511997340128173, −18.11019288222789423813422832816, −17.39786454397181978395969178858, −16.84517187367220203195858684828, −15.62701115205615251952823763476, −15.112454009485232202326818694465, −14.44682404546017255320541718248, −13.65790236457375948057373067808, −12.86874612299362720066114045738, −11.69274307012615707641078895324, −10.790383025521005531800495610276, −9.77592535445556678461154348205, −9.35511066202501234166087706436, −8.36435707817036066410926107152, −7.54120326930205754808649223638, −6.58018087562777314080606650791, −5.94631801343904299304961181062, −5.124909919841640391585400021296, −4.27449998050452576781157703975, −2.54623477781625515801041592360, −2.01468038387319622652404362437, −0.39235509276198605577877982652, 0.786405571323682702963878500104, 1.78490162105601966387743562501, 2.41485973243012746779896974859, 3.89554415605957385756961957488, 4.46472754808796497057720099558, 5.42012992785983234039435148903, 6.65086904991314581726444967593, 7.71704835120197365281808116362, 8.49237636200661398681703733416, 9.33220604338464829045264873370, 10.15365136906941986320610953672, 10.65861651955009847042375069350, 11.609592134388119351075026420437, 12.73153089891259101482322758202, 12.931717718490341761579707061664, 14.14901262628926452746610227638, 14.44131876114023849823005282214, 16.07428443657123120595717885643, 16.86266241593570737616689026989, 17.49629897284416948524186147997, 17.848898251420871456481045128154, 19.071239875787480592279616593187, 19.77571739647693727421124298721, 20.38001590481622358888376831716, 21.10449095100977900666696587049

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