Properties

Label 1-33e2-1089.254-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.629 + 0.777i$
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  + (0.995 − 0.0950i)2-s + (0.981 − 0.189i)4-s + (−0.723 + 0.690i)5-s + (−0.786 + 0.618i)7-s + (0.959 − 0.281i)8-s + (−0.654 + 0.755i)10-s + (0.981 − 0.189i)13-s + (−0.723 + 0.690i)14-s + (0.928 − 0.371i)16-s + (−0.841 − 0.540i)17-s + (0.841 − 0.540i)19-s + (−0.580 + 0.814i)20-s + (0.786 + 0.618i)23-s + (0.0475 − 0.998i)25-s + (0.959 − 0.281i)26-s + ⋯
L(s)  = 1  + (0.995 − 0.0950i)2-s + (0.981 − 0.189i)4-s + (−0.723 + 0.690i)5-s + (−0.786 + 0.618i)7-s + (0.959 − 0.281i)8-s + (−0.654 + 0.755i)10-s + (0.981 − 0.189i)13-s + (−0.723 + 0.690i)14-s + (0.928 − 0.371i)16-s + (−0.841 − 0.540i)17-s + (0.841 − 0.540i)19-s + (−0.580 + 0.814i)20-s + (0.786 + 0.618i)23-s + (0.0475 − 0.998i)25-s + (0.959 − 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.629 + 0.777i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.629 + 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.271441008 + 1.560107710i\)
\(L(\frac12)\) \(\approx\) \(3.271441008 + 1.560107710i\)
\(L(1)\) \(\approx\) \(1.769128479 + 0.2471459752i\)
\(L(1)\) \(\approx\) \(1.769128479 + 0.2471459752i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.995 - 0.0950i)T \)
5 \( 1 + (-0.723 + 0.690i)T \)
7 \( 1 + (-0.786 + 0.618i)T \)
13 \( 1 + (0.981 - 0.189i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (0.786 + 0.618i)T \)
29 \( 1 + (0.888 + 0.458i)T \)
31 \( 1 + (-0.327 - 0.945i)T \)
37 \( 1 + (-0.654 + 0.755i)T \)
41 \( 1 + (-0.580 - 0.814i)T \)
43 \( 1 + (0.723 + 0.690i)T \)
47 \( 1 + (-0.580 + 0.814i)T \)
53 \( 1 + (0.142 + 0.989i)T \)
59 \( 1 + (0.995 + 0.0950i)T \)
61 \( 1 + (0.580 - 0.814i)T \)
67 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (-0.142 + 0.989i)T \)
79 \( 1 + (0.235 + 0.971i)T \)
83 \( 1 + (0.786 - 0.618i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (0.723 + 0.690i)T \)
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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

−20.94304572508477435553922343640, −20.58892713201389214657111351890, −19.62040057036793659763863420735, −19.30217663845208586650927957441, −17.93670269755190495317249350124, −16.84214899864553217715649884045, −16.21877769576419325316449491761, −15.7905049386895258376160597459, −14.90585508969185741029171177390, −13.86932353442129545966023384446, −13.244159388716663561375981936653, −12.6014823523865531861379304299, −11.82358660212780861375274506213, −10.97393294058011760436857517280, −10.227857145796605869257261380999, −8.91741420203454486507310406797, −8.1611754987375419091536833230, −7.10060958404482793110490866375, −6.51282426668829885619735022169, −5.46836196836069175226387395702, −4.53130132809181947839579565832, −3.771273536309019525003012491324, −3.17134416476620670733281856909, −1.70444143258370405051093170858, −0.61993594321354877085234165263, 0.93306903846178468070750507194, 2.4575279389270911687162587456, 3.11453838127411046406712989073, 3.8138191406634121757627561720, 4.89306243594224476536763353739, 5.838714431470338681054251134717, 6.69742067626145576094072084915, 7.23325686291279330483678116328, 8.39191867388689110721962129106, 9.44301783806023546190825139076, 10.495191338739115384143996929165, 11.33606321966611135102921298009, 11.75411482227340005564010983444, 12.79991332469734742378771550240, 13.43364010380192494285146451336, 14.2480184882799761116630866050, 15.23045716365359840363303133541, 15.78161400829306027779793383991, 16.08998921844216438268884734844, 17.48526566703859882107184802948, 18.533862211438293493781281694612, 19.115535396570958939053182461143, 19.924374706373687041016586171977, 20.56322873704734667622146056391, 21.584823636311649146413563561991

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