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Properties

Label 1-33e2-1089.185-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.656 + 0.754i$
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  + (0.761 − 0.647i)2-s + (0.161 − 0.986i)4-s + (0.290 + 0.956i)5-s + (−0.217 − 0.976i)7-s + (−0.516 − 0.856i)8-s + (0.841 + 0.540i)10-s + (0.449 + 0.893i)13-s + (−0.797 − 0.603i)14-s + (−0.948 − 0.318i)16-s + (0.466 + 0.884i)17-s + (−0.985 − 0.170i)19-s + (0.991 − 0.132i)20-s + (0.995 − 0.0950i)23-s + (−0.830 + 0.556i)25-s + (0.921 + 0.389i)26-s + ⋯
L(s)  = 1  + (0.761 − 0.647i)2-s + (0.161 − 0.986i)4-s + (0.290 + 0.956i)5-s + (−0.217 − 0.976i)7-s + (−0.516 − 0.856i)8-s + (0.841 + 0.540i)10-s + (0.449 + 0.893i)13-s + (−0.797 − 0.603i)14-s + (−0.948 − 0.318i)16-s + (0.466 + 0.884i)17-s + (−0.985 − 0.170i)19-s + (0.991 − 0.132i)20-s + (0.995 − 0.0950i)23-s + (−0.830 + 0.556i)25-s + (0.921 + 0.389i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.656 + 0.754i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.656 + 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.001041529 + 0.9114043452i\)
\(L(\frac12)\) \(\approx\) \(2.001041529 + 0.9114043452i\)
\(L(1)\) \(\approx\) \(1.440768036 - 0.3169716538i\)
\(L(1)\) \(\approx\) \(1.440768036 - 0.3169716538i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.761 - 0.647i)T \)
5 \( 1 + (0.290 + 0.956i)T \)
7 \( 1 + (-0.217 - 0.976i)T \)
13 \( 1 + (0.449 + 0.893i)T \)
17 \( 1 + (0.466 + 0.884i)T \)
19 \( 1 + (-0.985 - 0.170i)T \)
23 \( 1 + (0.995 - 0.0950i)T \)
29 \( 1 + (0.0665 - 0.997i)T \)
31 \( 1 + (-0.625 + 0.780i)T \)
37 \( 1 + (-0.254 + 0.967i)T \)
41 \( 1 + (-0.879 + 0.475i)T \)
43 \( 1 + (-0.327 - 0.945i)T \)
47 \( 1 + (0.432 + 0.901i)T \)
53 \( 1 + (-0.198 + 0.980i)T \)
59 \( 1 + (0.851 - 0.524i)T \)
61 \( 1 + (-0.179 + 0.983i)T \)
67 \( 1 + (0.723 + 0.690i)T \)
71 \( 1 + (-0.696 - 0.717i)T \)
73 \( 1 + (0.993 + 0.113i)T \)
79 \( 1 + (-0.905 - 0.424i)T \)
83 \( 1 + (-0.861 + 0.508i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (-0.290 + 0.956i)T \)
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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

−21.2090408318764120608357815781, −20.66243437067201632088042432208, −19.819385009131896837945320539595, −18.616946014564931920009945005443, −17.89963597579338402577075485947, −17.00270590307613592928427789100, −16.36310588554226109397086820558, −15.63991777278713523545512362889, −14.97256503618817743572911139310, −14.10975653653671485350235510963, −12.96903819517478188749311068543, −12.84557990220972580357068219715, −11.94886544353287220184665778291, −11.05532239036678789142060307913, −9.72769909046986689295206422900, −8.79055405221869539844342322149, −8.368424261766793473367555225602, −7.27504964129760386668707139163, −6.260293806420793859441284740, −5.390481629918830223870338667196, −5.0818774404448235060222749369, −3.82465437716031088131986671853, −2.90409958641871650621703289188, −1.867949635748259946229675251615, −0.329338474337013569080611381917, 1.16292429211352158584359633978, 2.06175887296577544117377046422, 3.157568330547038662939931866685, 3.84744637217178651625562749976, 4.6608797665521698313187892020, 5.92596051771938909806361735806, 6.59844138913691672778046801257, 7.22501797587722282076737489892, 8.63312946362717999360659333811, 9.74695227533885518572036366665, 10.44677212423069383903889663571, 10.95159880595199120806946155930, 11.75786157041857329365414529843, 12.852785632345482237163008993914, 13.50264496774350298748638834479, 14.17236244966412517184598011176, 14.84447224468976786544150477069, 15.594883788162979739680828397174, 16.76639273691173109636091985899, 17.425155056882614326382499542275, 18.70478689579548105153693431152, 19.018308729902021256132052486785, 19.79969698124400117000456366576, 20.79272889409418597589038727065, 21.35656770439748810894736599058

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