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Properties

Label 1-1083-1083.392-r0-0-0
Degree $1$
Conductor $1083$
Sign $0.868 - 0.494i$
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  + (−0.754 + 0.656i)2-s + (0.137 − 0.990i)4-s + (0.962 + 0.272i)5-s + (0.789 − 0.614i)7-s + (0.546 + 0.837i)8-s + (−0.904 + 0.426i)10-s + (0.677 − 0.735i)11-s + (−0.993 + 0.110i)13-s + (−0.191 + 0.981i)14-s + (−0.962 − 0.272i)16-s + (−0.137 − 0.990i)17-s + (0.401 − 0.915i)20-s + (−0.0275 + 0.999i)22-s + (0.592 − 0.805i)23-s + (0.851 + 0.523i)25-s + (0.677 − 0.735i)26-s + ⋯
L(s)  = 1  + (−0.754 + 0.656i)2-s + (0.137 − 0.990i)4-s + (0.962 + 0.272i)5-s + (0.789 − 0.614i)7-s + (0.546 + 0.837i)8-s + (−0.904 + 0.426i)10-s + (0.677 − 0.735i)11-s + (−0.993 + 0.110i)13-s + (−0.191 + 0.981i)14-s + (−0.962 − 0.272i)16-s + (−0.137 − 0.990i)17-s + (0.401 − 0.915i)20-s + (−0.0275 + 0.999i)22-s + (0.592 − 0.805i)23-s + (0.851 + 0.523i)25-s + (0.677 − 0.735i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1083\)    =    \(3 \cdot 19^{2}\)
Sign: $0.868 - 0.494i$
Analytic conductor: \(5.02943\)
Root analytic conductor: \(5.02943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1083} (392, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1083,\ (0:\ ),\ 0.868 - 0.494i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.248258462 - 0.3305918973i\)
\(L(\frac12)\) \(\approx\) \(1.248258462 - 0.3305918973i\)
\(L(1)\) \(\approx\) \(0.9563478804 + 0.03974975348i\)
\(L(1)\) \(\approx\) \(0.9563478804 + 0.03974975348i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.754 + 0.656i)T \)
5 \( 1 + (0.962 + 0.272i)T \)
7 \( 1 + (0.789 - 0.614i)T \)
11 \( 1 + (0.677 - 0.735i)T \)
13 \( 1 + (-0.993 + 0.110i)T \)
17 \( 1 + (-0.137 - 0.990i)T \)
23 \( 1 + (0.592 - 0.805i)T \)
29 \( 1 + (-0.926 - 0.376i)T \)
31 \( 1 + (-0.945 + 0.324i)T \)
37 \( 1 + (0.677 - 0.735i)T \)
41 \( 1 + (0.635 - 0.771i)T \)
43 \( 1 + (0.904 + 0.426i)T \)
47 \( 1 + (-0.975 - 0.218i)T \)
53 \( 1 + (0.975 + 0.218i)T \)
59 \( 1 + (0.635 - 0.771i)T \)
61 \( 1 + (-0.998 - 0.0550i)T \)
67 \( 1 + (-0.451 + 0.892i)T \)
71 \( 1 + (-0.998 + 0.0550i)T \)
73 \( 1 + (0.137 + 0.990i)T \)
79 \( 1 + (-0.904 - 0.426i)T \)
83 \( 1 + (-0.245 + 0.969i)T \)
89 \( 1 + (0.137 - 0.990i)T \)
97 \( 1 + (-0.451 - 0.892i)T \)
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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

−21.512607513298405853829313151436, −20.695039470087529940098127611281, −19.97171076831743695411353947736, −19.24812220472559466700409834637, −18.24623781795190121646699687842, −17.6810738593240394732165068960, −17.12550867024798985268713339519, −16.4922006985181587918086681870, −15.039836214006331455654578238524, −14.66488476406324742647985671672, −13.36278109694985069040525177967, −12.68375535962308718249444663287, −11.97909609940461713026943140926, −11.13548367277167161083995479138, −10.27327232570479347220025790953, −9.362917715522387316927362802633, −9.05556921488583304894248590170, −7.93908191497112771052735266369, −7.16714739100847123643481504271, −6.03053965956363608750000265779, −5.023652957467466704076016736, −4.13172547266329026968520618991, −2.7967342175974281785234233211, −1.90376228727122790184722987322, −1.38652809862449326367025927822, 0.71210819851817118861527601485, 1.7726682711077628007920444955, 2.69010801974885095847359176573, 4.294679331079186914014697583178, 5.23500080858048047053722081463, 5.95095402914133595904373905617, 7.042533541920112056563277224340, 7.41100806469093692001046614785, 8.66233641852528488371059287309, 9.292233682691371084095633634845, 10.054298544018046644106377793393, 10.95588462856638320341763151992, 11.46469682043383719101676784972, 12.91222952702903930536021482323, 13.95447412927724179958869619311, 14.38293511716838672097613657549, 14.9331468503560783381546290750, 16.30134449376196013388795295232, 16.79148893918808860287950914398, 17.47758926000155401008449647582, 18.1017084955047247213850614223, 18.862673300756088865742634034677, 19.72496475515604843165696701657, 20.52138560613865414986365906465, 21.28722971316759881068774892053

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