Properties

Label 1-33e2-1089.52-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.600 - 0.799i$
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  + (−0.797 − 0.603i)2-s + (0.272 + 0.962i)4-s + (−0.710 − 0.703i)5-s + (0.179 + 0.983i)7-s + (0.362 − 0.931i)8-s + (0.142 + 0.989i)10-s + (0.830 − 0.556i)13-s + (0.449 − 0.893i)14-s + (−0.851 + 0.524i)16-s + (0.736 − 0.676i)17-s + (−0.198 − 0.980i)19-s + (0.483 − 0.875i)20-s + (0.723 − 0.690i)23-s + (0.00951 + 0.999i)25-s + (−0.998 − 0.0570i)26-s + ⋯
L(s)  = 1  + (−0.797 − 0.603i)2-s + (0.272 + 0.962i)4-s + (−0.710 − 0.703i)5-s + (0.179 + 0.983i)7-s + (0.362 − 0.931i)8-s + (0.142 + 0.989i)10-s + (0.830 − 0.556i)13-s + (0.449 − 0.893i)14-s + (−0.851 + 0.524i)16-s + (0.736 − 0.676i)17-s + (−0.198 − 0.980i)19-s + (0.483 − 0.875i)20-s + (0.723 − 0.690i)23-s + (0.00951 + 0.999i)25-s + (−0.998 − 0.0570i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.600 - 0.799i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.600 - 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4786711238 - 0.9585071925i\)
\(L(\frac12)\) \(\approx\) \(0.4786711238 - 0.9585071925i\)
\(L(1)\) \(\approx\) \(0.6576367921 - 0.2767230374i\)
\(L(1)\) \(\approx\) \(0.6576367921 - 0.2767230374i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.797 - 0.603i)T \)
5 \( 1 + (-0.710 - 0.703i)T \)
7 \( 1 + (0.179 + 0.983i)T \)
13 \( 1 + (0.830 - 0.556i)T \)
17 \( 1 + (0.736 - 0.676i)T \)
19 \( 1 + (-0.198 - 0.980i)T \)
23 \( 1 + (0.723 - 0.690i)T \)
29 \( 1 + (-0.861 - 0.508i)T \)
31 \( 1 + (0.640 - 0.768i)T \)
37 \( 1 + (-0.466 + 0.884i)T \)
41 \( 1 + (0.683 - 0.730i)T \)
43 \( 1 + (0.888 + 0.458i)T \)
47 \( 1 + (-0.905 + 0.424i)T \)
53 \( 1 + (-0.0285 + 0.999i)T \)
59 \( 1 + (-0.290 + 0.956i)T \)
61 \( 1 + (-0.123 - 0.992i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (0.993 + 0.113i)T \)
73 \( 1 + (-0.610 - 0.791i)T \)
79 \( 1 + (0.935 + 0.353i)T \)
83 \( 1 + (0.432 - 0.901i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (-0.710 + 0.703i)T \)
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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

−21.28022050507378766417338186230, −20.63899563200951325087331606110, −19.57440403547932270907532832760, −19.20674633416587078239662625185, −18.407230828463228986127193401825, −17.64811674852095066711958934743, −16.761344574590210986426355838072, −16.21102853327381357580177876116, −15.38655717005422523978380429791, −14.43327700682021903477292156173, −14.16199138370010253916705641885, −12.92878797570531495690728508007, −11.68311859547413108146127492162, −10.93221954496304849779854598639, −10.45148933044148690807204649517, −9.53009676007247792813427899989, −8.41719529523212951808610230140, −7.80770879233304986048848585814, −7.051552527792641519786424437896, −6.37766367904111003391323489367, −5.33983480680323365666655947230, −4.080720619638700651405278592018, −3.38327543106721558393570849666, −1.80447322121933528435719497249, −0.9208578244184598291644712084, 0.39043917524746764654955239596, 1.19140602798548759297393430408, 2.46900786068657443427827675741, 3.25395242511585993742308516862, 4.34002169308624210228664995540, 5.26941661659879739200739989086, 6.42568479970897635028184900289, 7.63549112064343407729726981908, 8.1893380167935723260819846209, 9.01761615219191239724827692337, 9.502363910177359604981809782434, 10.82207914632229628807875474175, 11.38321666628131447041085811814, 12.16719979002831109925351232585, 12.77415388405544277954118141775, 13.58894112801808453342967418334, 15.0849015227085276460299532233, 15.6272170497438292311104415415, 16.36318661635080668270880532506, 17.18008156531052880761778024877, 17.98492408494164493718336181785, 18.890935741447464209701222219966, 19.157959968750714355246411999492, 20.34882266641180285072835703471, 20.73055382126785628254289054710

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