Properties

Label 1-33e2-1089.335-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.999 + 0.0175i$
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  + (−0.00951 − 0.999i)2-s + (−0.999 + 0.0190i)4-s + (0.761 − 0.647i)5-s + (−0.0665 − 0.997i)7-s + (0.0285 + 0.999i)8-s + (−0.654 − 0.755i)10-s + (0.820 + 0.572i)13-s + (−0.997 + 0.0760i)14-s + (0.999 − 0.0380i)16-s + (0.998 + 0.0570i)17-s + (−0.254 + 0.967i)19-s + (−0.749 + 0.662i)20-s + (−0.928 − 0.371i)23-s + (0.161 − 0.986i)25-s + (0.564 − 0.825i)26-s + ⋯
L(s)  = 1  + (−0.00951 − 0.999i)2-s + (−0.999 + 0.0190i)4-s + (0.761 − 0.647i)5-s + (−0.0665 − 0.997i)7-s + (0.0285 + 0.999i)8-s + (−0.654 − 0.755i)10-s + (0.820 + 0.572i)13-s + (−0.997 + 0.0760i)14-s + (0.999 − 0.0380i)16-s + (0.998 + 0.0570i)17-s + (−0.254 + 0.967i)19-s + (−0.749 + 0.662i)20-s + (−0.928 − 0.371i)23-s + (0.161 − 0.986i)25-s + (0.564 − 0.825i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.0175i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.999 + 0.0175i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.542856109 + 0.01357558222i\)
\(L(\frac12)\) \(\approx\) \(1.542856109 + 0.01357558222i\)
\(L(1)\) \(\approx\) \(0.9322976524 - 0.5077988719i\)
\(L(1)\) \(\approx\) \(0.9322976524 - 0.5077988719i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.00951 - 0.999i)T \)
5 \( 1 + (0.761 - 0.647i)T \)
7 \( 1 + (-0.0665 - 0.997i)T \)
13 \( 1 + (0.820 + 0.572i)T \)
17 \( 1 + (0.998 + 0.0570i)T \)
19 \( 1 + (-0.254 + 0.967i)T \)
23 \( 1 + (-0.928 - 0.371i)T \)
29 \( 1 + (0.935 + 0.353i)T \)
31 \( 1 + (-0.683 + 0.730i)T \)
37 \( 1 + (-0.921 + 0.389i)T \)
41 \( 1 + (0.217 + 0.976i)T \)
43 \( 1 + (0.235 + 0.971i)T \)
47 \( 1 + (0.398 + 0.917i)T \)
53 \( 1 + (0.466 + 0.884i)T \)
59 \( 1 + (-0.953 + 0.299i)T \)
61 \( 1 + (0.861 + 0.508i)T \)
67 \( 1 + (-0.995 - 0.0950i)T \)
71 \( 1 + (0.362 + 0.931i)T \)
73 \( 1 + (-0.985 - 0.170i)T \)
79 \( 1 + (-0.991 - 0.132i)T \)
83 \( 1 + (0.969 + 0.244i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (-0.761 - 0.647i)T \)
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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

−21.561988800387514816078516764137, −20.6852624632955071401029571475, −19.28153948987099405253713047749, −18.71702838529928058819046018480, −17.92847750205869431420780048843, −17.5535824818934405307929534352, −16.49410350837247648744536255305, −15.59570771907346856639065093397, −15.17106163240969203954661615334, −14.21503123356208417388423395840, −13.63629006820906380492075743101, −12.79110113681493294455497796258, −11.88420707300653302234688917380, −10.665592552622978508137392120520, −9.90714176011745535726780033373, −9.076573378212152548258685307779, −8.37690981190490421410977482307, −7.38486241821967107400845495705, −6.5069673778565570410502608984, −5.68977012766838006591995601600, −5.34749228626832706401538788054, −3.89054223407615240411958670361, −2.95641359802239599729276215687, −1.81048751624384159534175016018, −0.33019798140872878453502768894, 1.203670766183185283325917590750, 1.438003013509303002469496836410, 2.83058404075049216222034715776, 3.8757675382528261423731999111, 4.51506854346145825033828126382, 5.57841071597034161312514217234, 6.41553164833078018633268308219, 7.81313442947371782902775451453, 8.5592383025303206529185315327, 9.44146097191248925119976032005, 10.24581803863618618575859219448, 10.6643468337799189864572014995, 11.86718770014888356844018270237, 12.52850632496554076254384419539, 13.29911082095008134033508143608, 14.10483159544529090085846840589, 14.38741939192679080319564537659, 16.201076486755325831887447951416, 16.61674168314396217038754380391, 17.51698101310180359792257436447, 18.18657380353901602355881758919, 19.01234332624173698465338586778, 19.90050547974197384646319019803, 20.54289405172590831677569775277, 21.12552008403739293846061140054

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