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Properties

Label 1-1083-1083.692-r0-0-0
Degree $1$
Conductor $1083$
Sign $-0.607 + 0.794i$
Analytic cond. $5.02943$
Root an. cond. $5.02943$
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  + (0.851 + 0.523i)2-s + (0.451 + 0.892i)4-s + (0.592 − 0.805i)5-s + (0.546 + 0.837i)7-s + (−0.0825 + 0.996i)8-s + (0.926 − 0.376i)10-s + (−0.945 + 0.324i)11-s + (−0.635 + 0.771i)13-s + (0.0275 + 0.999i)14-s + (−0.592 + 0.805i)16-s + (−0.451 + 0.892i)17-s + (0.986 + 0.164i)20-s + (−0.975 − 0.218i)22-s + (−0.350 + 0.936i)23-s + (−0.298 − 0.954i)25-s + (−0.945 + 0.324i)26-s + ⋯
L(s)  = 1  + (0.851 + 0.523i)2-s + (0.451 + 0.892i)4-s + (0.592 − 0.805i)5-s + (0.546 + 0.837i)7-s + (−0.0825 + 0.996i)8-s + (0.926 − 0.376i)10-s + (−0.945 + 0.324i)11-s + (−0.635 + 0.771i)13-s + (0.0275 + 0.999i)14-s + (−0.592 + 0.805i)16-s + (−0.451 + 0.892i)17-s + (0.986 + 0.164i)20-s + (−0.975 − 0.218i)22-s + (−0.350 + 0.936i)23-s + (−0.298 − 0.954i)25-s + (−0.945 + 0.324i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1083\)    =    \(3 \cdot 19^{2}\)
Sign: $-0.607 + 0.794i$
Analytic conductor: \(5.02943\)
Root analytic conductor: \(5.02943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1083} (692, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1083,\ (0:\ ),\ -0.607 + 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.031623817 + 2.087154344i\)
\(L(\frac12)\) \(\approx\) \(1.031623817 + 2.087154344i\)
\(L(1)\) \(\approx\) \(1.454514097 + 0.8731115846i\)
\(L(1)\) \(\approx\) \(1.454514097 + 0.8731115846i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.851 + 0.523i)T \)
5 \( 1 + (0.592 - 0.805i)T \)
7 \( 1 + (0.546 + 0.837i)T \)
11 \( 1 + (-0.945 + 0.324i)T \)
13 \( 1 + (-0.635 + 0.771i)T \)
17 \( 1 + (-0.451 + 0.892i)T \)
23 \( 1 + (-0.350 + 0.936i)T \)
29 \( 1 + (-0.998 + 0.0550i)T \)
31 \( 1 + (0.879 + 0.475i)T \)
37 \( 1 + (-0.945 + 0.324i)T \)
41 \( 1 + (0.716 - 0.697i)T \)
43 \( 1 + (-0.926 - 0.376i)T \)
47 \( 1 + (0.191 - 0.981i)T \)
53 \( 1 + (-0.191 + 0.981i)T \)
59 \( 1 + (0.716 - 0.697i)T \)
61 \( 1 + (0.904 + 0.426i)T \)
67 \( 1 + (0.821 + 0.569i)T \)
71 \( 1 + (0.904 - 0.426i)T \)
73 \( 1 + (0.451 - 0.892i)T \)
79 \( 1 + (0.926 + 0.376i)T \)
83 \( 1 + (0.401 - 0.915i)T \)
89 \( 1 + (0.451 + 0.892i)T \)
97 \( 1 + (0.821 - 0.569i)T \)
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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

−21.05857516701514026874105697285, −20.6556417557819937857633364843, −19.84931216766278498487354831647, −18.88637266515874722405148935830, −18.17587529129709847665140931361, −17.472389740924978477456426363447, −16.36104049036423831358156247416, −15.4395787480449325684871236351, −14.666012677173455945227864443, −14.04383309187630542551269591052, −13.360257960062664680783574246127, −12.709144948793663464349766942332, −11.47053410733662937029034177292, −10.927518377865259891237384629932, −10.210424057848943018547458864033, −9.638384382310658921491743318120, −8.07101502691387325938140294320, −7.221451905905461304401813364084, −6.42647326199234312682491139950, −5.39439680121698859519664392985, −4.78551428677306237344114665924, −3.65150628152306728070269656338, −2.70817664412959967603794563469, −2.09573476818523355418273369511, −0.6454667919209451064049650905, 1.89220601733666521570806992027, 2.26318681694684404366235743484, 3.69994239855435664726128390167, 4.78151714398954359323949484357, 5.25160719296625839459582561744, 6.002508429984880640407118329110, 7.03685450553871770296800713792, 8.04823566885631289707441821358, 8.67150720612788955697360042259, 9.58744268264396445038830887051, 10.73053582984279621197673213752, 11.84155700005014591594599690778, 12.3252432194043262865305496808, 13.16226861495863643108681579374, 13.80544882448812468650133546351, 14.71902173280992828251364993121, 15.454357715356545797175403073634, 16.05849741570778266962210700749, 17.14845844588080568899537217373, 17.480960169787217481247025846466, 18.44429065353406303990707766609, 19.56214774942679714024874253634, 20.54900600733743644166953996506, 21.15732516056515775158960671053, 21.68750973156941789225002312375

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