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Properties

Label 1-33e2-1089.113-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.336 + 0.941i$
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  + (−0.640 + 0.768i)2-s + (−0.179 − 0.983i)4-s + (0.398 + 0.917i)5-s + (0.988 + 0.151i)7-s + (0.870 + 0.491i)8-s + (−0.959 − 0.281i)10-s + (0.879 − 0.475i)13-s + (−0.749 + 0.662i)14-s + (−0.935 + 0.353i)16-s + (−0.516 + 0.856i)17-s + (0.0855 − 0.996i)19-s + (0.830 − 0.556i)20-s + (0.888 + 0.458i)23-s + (−0.683 + 0.730i)25-s + (−0.198 + 0.980i)26-s + ⋯
L(s)  = 1  + (−0.640 + 0.768i)2-s + (−0.179 − 0.983i)4-s + (0.398 + 0.917i)5-s + (0.988 + 0.151i)7-s + (0.870 + 0.491i)8-s + (−0.959 − 0.281i)10-s + (0.879 − 0.475i)13-s + (−0.749 + 0.662i)14-s + (−0.935 + 0.353i)16-s + (−0.516 + 0.856i)17-s + (0.0855 − 0.996i)19-s + (0.830 − 0.556i)20-s + (0.888 + 0.458i)23-s + (−0.683 + 0.730i)25-s + (−0.198 + 0.980i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.336 + 0.941i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.336 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.167397895 + 1.657022290i\)
\(L(\frac12)\) \(\approx\) \(1.167397895 + 1.657022290i\)
\(L(1)\) \(\approx\) \(0.8927001230 + 0.5083822668i\)
\(L(1)\) \(\approx\) \(0.8927001230 + 0.5083822668i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.640 + 0.768i)T \)
5 \( 1 + (0.398 + 0.917i)T \)
7 \( 1 + (0.988 + 0.151i)T \)
13 \( 1 + (0.879 - 0.475i)T \)
17 \( 1 + (-0.516 + 0.856i)T \)
19 \( 1 + (0.0855 - 0.996i)T \)
23 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (0.290 - 0.956i)T \)
31 \( 1 + (0.997 + 0.0760i)T \)
37 \( 1 + (0.610 + 0.791i)T \)
41 \( 1 + (-0.272 + 0.962i)T \)
43 \( 1 + (-0.995 - 0.0950i)T \)
47 \( 1 + (-0.999 - 0.0380i)T \)
53 \( 1 + (-0.774 + 0.633i)T \)
59 \( 1 + (0.969 + 0.244i)T \)
61 \( 1 + (0.345 - 0.938i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (0.921 + 0.389i)T \)
73 \( 1 + (-0.998 - 0.0570i)T \)
79 \( 1 + (0.953 - 0.299i)T \)
83 \( 1 + (0.710 + 0.703i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (-0.398 + 0.917i)T \)
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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

−20.93967467071374219564446908827, −20.419588074885068444694251241698, −19.5661822746596479036266643533, −18.54217093313663553320376106267, −17.99946860721243752840767546820, −17.2759834158616498431365597692, −16.48102075422337627753670262443, −15.95008154778550991027958759706, −14.54185510749801125392957341834, −13.700308915130496094210017394238, −13.06478468675623275149010948647, −12.11128427203106026546189501361, −11.47473284171231714901525842446, −10.68625846729858491638930868356, −9.80831750650665739939379459689, −8.84762435229008611853363215125, −8.48693149281952493033723094831, −7.54702114492958787371301164335, −6.4555592038876288284972168528, −5.10438938239262675167837494218, −4.50434701554786428839048155575, −3.47898626341395532464417012472, −2.181657248969108604966259081139, −1.43175692367670010859441987732, −0.62961457753532762221228361774, 0.9684485375933882501660564950, 1.89963616055498392249843235451, 2.99936742919214104586555681781, 4.4200947525976949303230742769, 5.2965881739937986054343164400, 6.26062060997487682785713958047, 6.78413494296740186476195889618, 7.92571661755497252747167727892, 8.3989445277520069171677956770, 9.426768100115376204164592808580, 10.287001932020991411509984344570, 11.064130822354530655276711148886, 11.51610020346511636164744875456, 13.3177196050093751895079064110, 13.642155619489830816327341720360, 14.88156204719650534722277774176, 15.06093743806284755493422340290, 15.88233639934855803743958254114, 17.170897579585054687452303866235, 17.56545996104597864773814559787, 18.19215954412933105054200184384, 18.92502994316992052884735986418, 19.6810952291329801574734772731, 20.68509919897295027394646092369, 21.53096308018537683432840159035

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