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Properties

Label 1-33e2-1089.41-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.329 - 0.944i$
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  + (0.964 + 0.263i)2-s + (0.861 + 0.508i)4-s + (−0.640 − 0.768i)5-s + (0.290 − 0.956i)7-s + (0.696 + 0.717i)8-s + (−0.415 − 0.909i)10-s + (0.217 − 0.976i)13-s + (0.532 − 0.846i)14-s + (0.483 + 0.875i)16-s + (−0.0285 − 0.999i)17-s + (−0.610 + 0.791i)19-s + (−0.161 − 0.986i)20-s + (0.327 − 0.945i)23-s + (−0.179 + 0.983i)25-s + (0.466 − 0.884i)26-s + ⋯
L(s)  = 1  + (0.964 + 0.263i)2-s + (0.861 + 0.508i)4-s + (−0.640 − 0.768i)5-s + (0.290 − 0.956i)7-s + (0.696 + 0.717i)8-s + (−0.415 − 0.909i)10-s + (0.217 − 0.976i)13-s + (0.532 − 0.846i)14-s + (0.483 + 0.875i)16-s + (−0.0285 − 0.999i)17-s + (−0.610 + 0.791i)19-s + (−0.161 − 0.986i)20-s + (0.327 − 0.945i)23-s + (−0.179 + 0.983i)25-s + (0.466 − 0.884i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.329 - 0.944i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.329 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.929700793 - 1.370780569i\)
\(L(\frac12)\) \(\approx\) \(1.929700793 - 1.370780569i\)
\(L(1)\) \(\approx\) \(1.671186595 - 0.3175789557i\)
\(L(1)\) \(\approx\) \(1.671186595 - 0.3175789557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.964 + 0.263i)T \)
5 \( 1 + (-0.640 - 0.768i)T \)
7 \( 1 + (0.290 - 0.956i)T \)
13 \( 1 + (0.217 - 0.976i)T \)
17 \( 1 + (-0.0285 - 0.999i)T \)
19 \( 1 + (-0.610 + 0.791i)T \)
23 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (-0.761 + 0.647i)T \)
31 \( 1 + (-0.595 - 0.803i)T \)
37 \( 1 + (0.198 - 0.980i)T \)
41 \( 1 + (0.988 - 0.151i)T \)
43 \( 1 + (-0.928 - 0.371i)T \)
47 \( 1 + (-0.449 + 0.893i)T \)
53 \( 1 + (-0.516 - 0.856i)T \)
59 \( 1 + (0.625 - 0.780i)T \)
61 \( 1 + (0.710 + 0.703i)T \)
67 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (0.564 + 0.825i)T \)
73 \( 1 + (-0.0855 - 0.996i)T \)
79 \( 1 + (0.830 - 0.556i)T \)
83 \( 1 + (0.797 - 0.603i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (0.640 - 0.768i)T \)
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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

−21.64003949751394679517761950171, −21.18897437937362854664670189129, −19.99498824105476066932180153495, −19.260782090006742186488883712045, −18.843205008864854700880783794914, −17.85624330464975099844510302839, −16.70617421213663560109854013080, −15.785578628431021815146862954172, −15.05786334291663221265022794683, −14.77110916521468315668230993185, −13.74303383657916468656680134152, −12.90915838701601389155044176100, −12.02194139959655396372226076728, −11.361802155598248452411434590189, −10.90773452718277572181320138242, −9.77438442432766773444038029077, −8.72239492377223634928105685390, −7.70896522168643243715687871458, −6.73887965144190066102503979393, −6.13520975884850628965229730080, −5.10489354354628252593213128978, −4.18095191256538264676036328238, −3.39363019461581856541441798346, −2.43148718730424187043481646708, −1.61582617772723468213393923643, 0.68388658312961832784926088681, 1.97521137932703349662964441459, 3.31306901080845553964468503768, 3.99501061128601949471457647591, 4.80739051917782814724007240873, 5.52163935622844265584520096725, 6.63898740991433353551521619910, 7.63026109175456777087163855593, 7.990210529182782805228284002047, 9.11948242817157590114474885072, 10.464167042230742195924655033948, 11.10047659357251419507535917617, 11.93337113203544490924450224003, 12.94956749086961344378385078128, 13.11445367527336055078587844921, 14.38982342767204037194988851311, 14.81374325246567674524442522796, 15.934390220762026287242954051839, 16.41129736565404708092821282138, 17.0759714583356676143886758527, 18.017666126904250352086182107508, 19.23324685514676302676492417783, 20.12429206093206669077139987309, 20.61638290133971731956736597187, 21.0072465420138831815701646321

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