Properties

Label 1-33e2-1089.1021-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: $\chi_{1089} (1021, \cdot )$
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

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

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