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Properties

Label 1-33e2-1089.313-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.682 - 0.731i$
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  + (−0.991 − 0.132i)2-s + (0.964 + 0.263i)4-s + (−0.905 − 0.424i)5-s + (−0.595 − 0.803i)7-s + (−0.921 − 0.389i)8-s + (0.841 + 0.540i)10-s + (−0.625 − 0.780i)13-s + (0.483 + 0.875i)14-s + (0.861 + 0.508i)16-s + (0.696 − 0.717i)17-s + (0.897 − 0.441i)19-s + (−0.761 − 0.647i)20-s + (0.580 + 0.814i)23-s + (0.640 + 0.768i)25-s + (0.516 + 0.856i)26-s + ⋯
L(s)  = 1  + (−0.991 − 0.132i)2-s + (0.964 + 0.263i)4-s + (−0.905 − 0.424i)5-s + (−0.595 − 0.803i)7-s + (−0.921 − 0.389i)8-s + (0.841 + 0.540i)10-s + (−0.625 − 0.780i)13-s + (0.483 + 0.875i)14-s + (0.861 + 0.508i)16-s + (0.696 − 0.717i)17-s + (0.897 − 0.441i)19-s + (−0.761 − 0.647i)20-s + (0.580 + 0.814i)23-s + (0.640 + 0.768i)25-s + (0.516 + 0.856i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.682 - 0.731i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.682 - 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2298574726 - 0.5289489337i\)
\(L(\frac12)\) \(\approx\) \(0.2298574726 - 0.5289489337i\)
\(L(1)\) \(\approx\) \(0.5261851762 - 0.2122127613i\)
\(L(1)\) \(\approx\) \(0.5261851762 - 0.2122127613i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.991 - 0.132i)T \)
5 \( 1 + (-0.905 - 0.424i)T \)
7 \( 1 + (-0.595 - 0.803i)T \)
13 \( 1 + (-0.625 - 0.780i)T \)
17 \( 1 + (0.696 - 0.717i)T \)
19 \( 1 + (0.897 - 0.441i)T \)
23 \( 1 + (0.580 + 0.814i)T \)
29 \( 1 + (0.345 + 0.938i)T \)
31 \( 1 + (0.449 - 0.893i)T \)
37 \( 1 + (0.774 - 0.633i)T \)
41 \( 1 + (0.997 - 0.0760i)T \)
43 \( 1 + (0.981 + 0.189i)T \)
47 \( 1 + (-0.851 + 0.524i)T \)
53 \( 1 + (-0.870 - 0.491i)T \)
59 \( 1 + (-0.432 - 0.901i)T \)
61 \( 1 + (0.380 - 0.924i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (-0.466 + 0.884i)T \)
73 \( 1 + (-0.736 - 0.676i)T \)
79 \( 1 + (-0.290 - 0.956i)T \)
83 \( 1 + (-0.948 + 0.318i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (-0.905 + 0.424i)T \)
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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

−21.59562623989881885606338978890, −20.84450622856669352422219953211, −19.778245854407860693530561214286, −19.23168107410429292682895266554, −18.78051849559687907021706382374, −18.03782018837582761106806883526, −16.975881820712732806505823825111, −16.265832716722545934498681311984, −15.68766124462122723707584135389, −14.85326329946346094557810588031, −14.289665122912400773878855330914, −12.69127704176005136610533123762, −12.0007026877556173956191166935, −11.485848196540505891266676689002, −10.4046115982309447772329012590, −9.74617539406866585676903629270, −8.85084212821762409715377697883, −8.08960587751034985596827518611, −7.29014540597485627942846619448, −6.50165000636714361360401453158, −5.69716127578976114160690682357, −4.37251583799854711788693048445, −3.11881240443133044375227927944, −2.51993776554923471821898763072, −1.1293533192142946607658353436, 0.4292321235238394055937656223, 1.185384816685101126413991344176, 2.88240642389831214143844783329, 3.381249996721185427399279789147, 4.61862869127057099301536903312, 5.71089646955419717199244940583, 7.00005584441805829834973876456, 7.52968655670165452153413656319, 8.09205295395691624093296397288, 9.42625052688721697438089918163, 9.661207040492341630925094108536, 10.8761249220120236301711239528, 11.40247040322791688041499908280, 12.41894115463718382054087234423, 12.93045535508185863121425414776, 14.1883158534876516331354924619, 15.20962557267869618252073286722, 16.01416351176804352305235795231, 16.39842828012178522329287523856, 17.32373425228354991422526472846, 17.953373232455795132818844695818, 19.13561792282998890386006037997, 19.44257814645214487434284925557, 20.34046203163405748171636732071, 20.6071157368387198345016758802

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