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Properties

Label 1-33e2-1089.106-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.336 + 0.941i$
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  + (−0.640 − 0.768i)2-s + (−0.179 + 0.983i)4-s + (−0.398 + 0.917i)5-s + (−0.988 + 0.151i)7-s + (0.870 − 0.491i)8-s + (0.959 − 0.281i)10-s + (−0.879 − 0.475i)13-s + (0.749 + 0.662i)14-s + (−0.935 − 0.353i)16-s + (−0.516 − 0.856i)17-s + (−0.0855 − 0.996i)19-s + (−0.830 − 0.556i)20-s + (−0.888 + 0.458i)23-s + (−0.683 − 0.730i)25-s + (0.198 + 0.980i)26-s + ⋯
L(s)  = 1  + (−0.640 − 0.768i)2-s + (−0.179 + 0.983i)4-s + (−0.398 + 0.917i)5-s + (−0.988 + 0.151i)7-s + (0.870 − 0.491i)8-s + (0.959 − 0.281i)10-s + (−0.879 − 0.475i)13-s + (0.749 + 0.662i)14-s + (−0.935 − 0.353i)16-s + (−0.516 − 0.856i)17-s + (−0.0855 − 0.996i)19-s + (−0.830 − 0.556i)20-s + (−0.888 + 0.458i)23-s + (−0.683 − 0.730i)25-s + (0.198 + 0.980i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.336 + 0.941i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.336 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2765012327 + 0.1947994055i\)
\(L(\frac12)\) \(\approx\) \(0.2765012327 + 0.1947994055i\)
\(L(1)\) \(\approx\) \(0.5219197583 - 0.1047132079i\)
\(L(1)\) \(\approx\) \(0.5219197583 - 0.1047132079i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.640 - 0.768i)T \)
5 \( 1 + (-0.398 + 0.917i)T \)
7 \( 1 + (-0.988 + 0.151i)T \)
13 \( 1 + (-0.879 - 0.475i)T \)
17 \( 1 + (-0.516 - 0.856i)T \)
19 \( 1 + (-0.0855 - 0.996i)T \)
23 \( 1 + (-0.888 + 0.458i)T \)
29 \( 1 + (0.290 + 0.956i)T \)
31 \( 1 + (0.997 - 0.0760i)T \)
37 \( 1 + (0.610 - 0.791i)T \)
41 \( 1 + (-0.272 - 0.962i)T \)
43 \( 1 + (0.995 - 0.0950i)T \)
47 \( 1 + (0.999 - 0.0380i)T \)
53 \( 1 + (0.774 + 0.633i)T \)
59 \( 1 + (-0.969 + 0.244i)T \)
61 \( 1 + (-0.345 - 0.938i)T \)
67 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (-0.921 + 0.389i)T \)
73 \( 1 + (0.998 - 0.0570i)T \)
79 \( 1 + (-0.953 - 0.299i)T \)
83 \( 1 + (0.710 - 0.703i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (-0.398 - 0.917i)T \)
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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

−20.90513363096960812064759919121, −19.97809085552735065645699549643, −19.45940760480292821037710235861, −18.90714703353720321358264982192, −17.82780262724598471490853504769, −16.81012658002489286434666702100, −16.69995742532779376510318508181, −15.73143570904950472395801414749, −15.14930665695643358168118121613, −14.11442023142077387700813966875, −13.2983099065835851795106600232, −12.426723986081095063990216445041, −11.65803573761480418030598908373, −10.298670903245982030135142406531, −9.84028803605152169000175757071, −8.956860221566830207607332139324, −8.191448108339501708722574429724, −7.48379775838560205608991221967, −6.387654136482756057906401742914, −5.87100164453067310109513561800, −4.58959994874383845529087688043, −4.05132957584660611892245732851, −2.43096723871447795090815562999, −1.24228842730974978255280311755, −0.15275893885602509656453871268, 0.602015594237599765125423772778, 2.37038278734509393801302144707, 2.785393668296910824615559885367, 3.69821104436118560258982700568, 4.69606071444664139440815808978, 6.1096262121361528534020401740, 7.16667999528388405425298152797, 7.50954612121538445444549302559, 8.80297283201279581625599903167, 9.50490149939994304867998921338, 10.299510140834686770039662282543, 10.92423452671454517524353239855, 11.901275904917500544059935335999, 12.41479109639523263496332763676, 13.43974407581914207742451884020, 14.147835514803700497229564524960, 15.502969825010893835403662536106, 15.80765445377087617767564146863, 16.92079646886941777846688597741, 17.77926929737283341369548287657, 18.34522971180681835324251920652, 19.22188953315751606613949560217, 19.73492319073102405395584316147, 20.25444103599885402248799516993, 21.550523506129983622045155971293

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