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Properties

Label 1-33e2-1089.16-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.293 - 0.955i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (−0.290 − 0.956i)2-s + (−0.830 + 0.556i)4-s + (0.988 + 0.151i)5-s + (0.879 − 0.475i)7-s + (0.774 + 0.633i)8-s + (−0.142 − 0.989i)10-s + (0.999 + 0.0380i)13-s + (−0.710 − 0.703i)14-s + (0.380 − 0.924i)16-s + (0.198 − 0.980i)17-s + (0.993 + 0.113i)19-s + (−0.905 + 0.424i)20-s + (0.723 − 0.690i)23-s + (0.953 + 0.299i)25-s + (−0.254 − 0.967i)26-s + ⋯
L(s)  = 1  + (−0.290 − 0.956i)2-s + (−0.830 + 0.556i)4-s + (0.988 + 0.151i)5-s + (0.879 − 0.475i)7-s + (0.774 + 0.633i)8-s + (−0.142 − 0.989i)10-s + (0.999 + 0.0380i)13-s + (−0.710 − 0.703i)14-s + (0.380 − 0.924i)16-s + (0.198 − 0.980i)17-s + (0.993 + 0.113i)19-s + (−0.905 + 0.424i)20-s + (0.723 − 0.690i)23-s + (0.953 + 0.299i)25-s + (−0.254 − 0.967i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.293 - 0.955i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.293 - 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.448520059 - 1.070394130i\)
\(L(\frac12)\) \(\approx\) \(1.448520059 - 1.070394130i\)
\(L(1)\) \(\approx\) \(1.103731525 - 0.5363169240i\)
\(L(1)\) \(\approx\) \(1.103731525 - 0.5363169240i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.290 - 0.956i)T \)
5 \( 1 + (0.988 + 0.151i)T \)
7 \( 1 + (0.879 - 0.475i)T \)
13 \( 1 + (0.999 + 0.0380i)T \)
17 \( 1 + (0.198 - 0.980i)T \)
19 \( 1 + (0.993 + 0.113i)T \)
23 \( 1 + (0.723 - 0.690i)T \)
29 \( 1 + (-0.217 + 0.976i)T \)
31 \( 1 + (-0.969 + 0.244i)T \)
37 \( 1 + (-0.985 - 0.170i)T \)
41 \( 1 + (0.483 + 0.875i)T \)
43 \( 1 + (-0.888 - 0.458i)T \)
47 \( 1 + (0.123 + 0.992i)T \)
53 \( 1 + (0.610 - 0.791i)T \)
59 \( 1 + (-0.999 + 0.0190i)T \)
61 \( 1 + (-0.683 - 0.730i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (-0.870 + 0.491i)T \)
73 \( 1 + (-0.564 + 0.825i)T \)
79 \( 1 + (0.548 + 0.836i)T \)
83 \( 1 + (-0.179 - 0.983i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.988 - 0.151i)T \)
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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

−21.5772468115373123689970496332, −21.01358791222386578917488468961, −20.0429936516246800137753896025, −18.85284872473993467564981706123, −18.3318297968786684268501734109, −17.58538034236748467080682541822, −17.08864768687556746325955381671, −16.19306900371698970214975748958, −15.26785870071928839139320814272, −14.731710926228176026408070925343, −13.69316094672761530687876581246, −13.406017544004657440391662597261, −12.23465133716646495406187013134, −11.05971429131444384193066990007, −10.319949613056482381587736797293, −9.29892858601203302112552057789, −8.78025468798251976430064117251, −7.937932282180005563761526149471, −7.03490973862263928892295909893, −5.85928026633672102496700383262, −5.613850802947156045458463862, −4.640549108210669944421875997273, −3.46881794041147705095333827800, −1.88802246930846417691844102250, −1.20477231743767612733788583961, 1.06838631072802722605708131659, 1.671401204690051723243167268339, 2.816296242517688842097665043376, 3.6414144901648490397761319805, 4.88313863378210396299623451068, 5.399641950929844674690559766000, 6.80201918588125979620651124757, 7.68194202346493949395800659277, 8.73372280569765759270944656158, 9.31288091706035555027719716777, 10.27653208307126545046125559632, 10.91926039138389446741233741500, 11.53110715290283970507595897222, 12.60596765521446184729758932908, 13.41555969429945817202660777216, 14.05817220991081966775710323942, 14.58314618760140979589038591992, 16.10981314221534252480570619462, 16.84113709395635997968533946833, 17.6667482036828646303900739030, 18.3246850588432756816393101092, 18.637263534305024946246481434448, 20.07155617435177919244184771826, 20.52024947932065245273221053816, 21.11755563841447842854499167182

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