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Properties

Label 1-33e2-1089.133-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.777 - 0.629i$
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  + (−0.995 + 0.0950i)2-s + (0.981 − 0.189i)4-s + (0.723 − 0.690i)5-s + (−0.786 + 0.618i)7-s + (−0.959 + 0.281i)8-s + (−0.654 + 0.755i)10-s + (0.981 − 0.189i)13-s + (0.723 − 0.690i)14-s + (0.928 − 0.371i)16-s + (0.841 + 0.540i)17-s + (0.841 − 0.540i)19-s + (0.580 − 0.814i)20-s + (−0.786 − 0.618i)23-s + (0.0475 − 0.998i)25-s + (−0.959 + 0.281i)26-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)2-s + (0.981 − 0.189i)4-s + (0.723 − 0.690i)5-s + (−0.786 + 0.618i)7-s + (−0.959 + 0.281i)8-s + (−0.654 + 0.755i)10-s + (0.981 − 0.189i)13-s + (0.723 − 0.690i)14-s + (0.928 − 0.371i)16-s + (0.841 + 0.540i)17-s + (0.841 − 0.540i)19-s + (0.580 − 0.814i)20-s + (−0.786 − 0.618i)23-s + (0.0475 − 0.998i)25-s + (−0.959 + 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.777 - 0.629i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.777 - 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9799884074 - 0.3471116387i\)
\(L(\frac12)\) \(\approx\) \(0.9799884074 - 0.3471116387i\)
\(L(1)\) \(\approx\) \(0.7938401248 - 0.08419584701i\)
\(L(1)\) \(\approx\) \(0.7938401248 - 0.08419584701i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.995 + 0.0950i)T \)
5 \( 1 + (0.723 - 0.690i)T \)
7 \( 1 + (-0.786 + 0.618i)T \)
13 \( 1 + (0.981 - 0.189i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-0.786 - 0.618i)T \)
29 \( 1 + (-0.888 - 0.458i)T \)
31 \( 1 + (-0.327 - 0.945i)T \)
37 \( 1 + (-0.654 + 0.755i)T \)
41 \( 1 + (0.580 + 0.814i)T \)
43 \( 1 + (0.723 + 0.690i)T \)
47 \( 1 + (0.580 - 0.814i)T \)
53 \( 1 + (-0.142 - 0.989i)T \)
59 \( 1 + (-0.995 - 0.0950i)T \)
61 \( 1 + (0.580 - 0.814i)T \)
67 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (-0.142 + 0.989i)T \)
79 \( 1 + (0.235 + 0.971i)T \)
83 \( 1 + (-0.786 + 0.618i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (0.723 + 0.690i)T \)
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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

−21.289139596114666913955994738978, −20.64282749400727371609968262918, −19.91699947925092518642975952456, −18.94243996969170778487929795523, −18.49785828846112387655037473988, −17.733712807557704355356675166678, −16.96856236058963358099026733922, −16.11812261020019971264005815175, −15.6880677916772447711816462083, −14.29562986823612624755439802465, −13.87005608393004643327766642950, −12.747387869250767289732037727629, −11.87430219836279780301449044968, −10.79793575503793803600453668497, −10.4096661554900387682875446992, −9.49043633061159107948382730330, −9.0199703737348324689443343909, −7.55722894446411985351020155104, −7.23550191377854597087838974214, −6.12718229859763765501773381415, −5.62698532101162737985721612556, −3.65934567074086884478124108616, −3.20823430671271818657065177648, −1.97557883327486637865109711829, −1.039033542641595385892099625490, 0.71894533812595058453144922754, 1.76850515773983233177104065268, 2.70710509750695521294360820464, 3.77671662963913165888827039271, 5.39329075031850812823317560642, 5.938446123940997714936683711923, 6.66274914495825254464905215660, 7.94156526660058653114013678054, 8.540891243348591869441623524915, 9.53190651520093360171709954074, 9.77263744413993700848440157979, 10.85767659610090339045542281050, 11.804392581594850387241666986574, 12.61843113798086316124411966968, 13.338402474867454707962720645811, 14.408776577314814715146473161678, 15.44740660776014136498447302960, 16.08791498668184355073787185831, 16.70052111115381479316897700400, 17.46329364256337063658042406407, 18.38458695844842545528916645946, 18.7524552538778149559254741982, 19.82413066027127942958844844172, 20.4518333760851902878835455756, 21.14597569261697197801622848090

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