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Properties

Label 1-1063-1063.1036-r0-0-0
Degree $1$
Conductor $1063$
Sign $0.839 + 0.543i$
Analytic cond. $4.93655$
Root an. cond. $4.93655$
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  + (0.969 − 0.245i)2-s + (−0.999 − 0.0177i)3-s + (0.879 − 0.476i)4-s + (0.574 + 0.818i)5-s + (−0.973 + 0.228i)6-s + (−0.5 + 0.866i)7-s + (0.734 − 0.678i)8-s + (0.999 + 0.0354i)9-s + (0.758 + 0.651i)10-s + (0.984 + 0.176i)11-s + (−0.887 + 0.461i)12-s + (0.802 + 0.596i)13-s + (−0.271 + 0.962i)14-s + (−0.560 − 0.828i)15-s + (0.545 − 0.838i)16-s + (−0.964 − 0.263i)17-s + ⋯
L(s)  = 1  + (0.969 − 0.245i)2-s + (−0.999 − 0.0177i)3-s + (0.879 − 0.476i)4-s + (0.574 + 0.818i)5-s + (−0.973 + 0.228i)6-s + (−0.5 + 0.866i)7-s + (0.734 − 0.678i)8-s + (0.999 + 0.0354i)9-s + (0.758 + 0.651i)10-s + (0.984 + 0.176i)11-s + (−0.887 + 0.461i)12-s + (0.802 + 0.596i)13-s + (−0.271 + 0.962i)14-s + (−0.560 − 0.828i)15-s + (0.545 − 0.838i)16-s + (−0.964 − 0.263i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1063\)
Sign: $0.839 + 0.543i$
Analytic conductor: \(4.93655\)
Root analytic conductor: \(4.93655\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1063} (1036, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1063,\ (0:\ ),\ 0.839 + 0.543i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.339575799 + 0.6906940541i\)
\(L(\frac12)\) \(\approx\) \(2.339575799 + 0.6906940541i\)
\(L(1)\) \(\approx\) \(1.652042397 + 0.1510883403i\)
\(L(1)\) \(\approx\) \(1.652042397 + 0.1510883403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1063 \( 1 \)
good2 \( 1 + (0.969 - 0.245i)T \)
3 \( 1 + (-0.999 - 0.0177i)T \)
5 \( 1 + (0.574 + 0.818i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.984 + 0.176i)T \)
13 \( 1 + (0.802 + 0.596i)T \)
17 \( 1 + (-0.964 - 0.263i)T \)
19 \( 1 + (0.220 - 0.975i)T \)
23 \( 1 + (0.823 + 0.567i)T \)
29 \( 1 + (0.631 - 0.775i)T \)
31 \( 1 + (-0.999 - 0.0177i)T \)
37 \( 1 + (-0.0266 + 0.999i)T \)
41 \( 1 + (-0.931 + 0.364i)T \)
43 \( 1 + (0.734 + 0.678i)T \)
47 \( 1 + (0.684 - 0.728i)T \)
53 \( 1 + (-0.903 + 0.429i)T \)
59 \( 1 + (0.977 + 0.211i)T \)
61 \( 1 + (-0.992 - 0.123i)T \)
67 \( 1 + (0.515 + 0.857i)T \)
71 \( 1 + (-0.339 + 0.940i)T \)
73 \( 1 + (0.603 + 0.797i)T \)
79 \( 1 + (0.453 - 0.891i)T \)
83 \( 1 + (-0.0620 - 0.998i)T \)
89 \( 1 + (-0.132 - 0.991i)T \)
97 \( 1 + (0.781 - 0.624i)T \)
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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

−21.63102766173169872063911609528, −20.722178873482919644873263029197, −20.24508399040290544065903088200, −19.28008902671070381424539551, −17.926799785230370134704707687490, −17.27970430181528461424346746347, −16.53272600677552077588877789473, −16.2450805848524011416865760078, −15.26440322552210729169551659256, −14.08147329337969175171511210524, −13.50594518094933112477503554123, −12.55356754515077861973229990376, −12.398160304816325673608783740003, −10.96179683157473024255845881041, −10.71355815658017561118041697330, −9.49269004423502160198073909733, −8.4555595646571934560141485984, −7.22172200416001159440141839110, −6.46928229522228129623218561864, −5.89250270782174236324401049893, −5.02857038083273428396240212997, −4.14931908251801831254914585194, −3.506692812125478078660782364, −1.819537480718320316847599568215, −0.93492068887701597976474568972, 1.32497571749412729810489778901, 2.26774694747575926246056392378, 3.266779807077313007924050874097, 4.28998642524175798179189847497, 5.22327597986931768716741989814, 6.105476796304780868696614485752, 6.59878732326611691837663233233, 7.131359245464077667026184427809, 9.053373912933058240730288817236, 9.72595697930951624070850344159, 10.74785297113079672863264847804, 11.5102319948355487551746110761, 11.781151710111645942959882754181, 13.05328368165182054794491305464, 13.428622428371986461555422099065, 14.46845908432225584032810597119, 15.40816075709689180264266666995, 15.78847803336475999411348026185, 16.87637472383894165841341236776, 17.653751548958677248249809050860, 18.59560922439431088959852208126, 19.11353719133380347691515806307, 20.10983560612429891511295430947, 21.286588870170997512178037618681, 21.81597630715987001946055315658

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