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Properties

Label 1-1063-1063.107-r1-0-0
Degree $1$
Conductor $1063$
Sign $0.717 + 0.696i$
Analytic cond. $114.235$
Root an. cond. $114.235$
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  + (−0.645 − 0.764i)2-s + (0.0620 + 0.998i)3-s + (−0.167 + 0.985i)4-s + (−0.977 + 0.211i)5-s + (0.722 − 0.691i)6-s + (−0.5 + 0.866i)7-s + (0.861 − 0.507i)8-s + (−0.992 + 0.123i)9-s + (0.792 + 0.610i)10-s + (−0.813 + 0.582i)11-s + (−0.994 − 0.106i)12-s + (−0.617 − 0.786i)13-s + (0.984 − 0.176i)14-s + (−0.271 − 0.962i)15-s + (−0.943 − 0.330i)16-s + (0.802 + 0.596i)17-s + ⋯
L(s)  = 1  + (−0.645 − 0.764i)2-s + (0.0620 + 0.998i)3-s + (−0.167 + 0.985i)4-s + (−0.977 + 0.211i)5-s + (0.722 − 0.691i)6-s + (−0.5 + 0.866i)7-s + (0.861 − 0.507i)8-s + (−0.992 + 0.123i)9-s + (0.792 + 0.610i)10-s + (−0.813 + 0.582i)11-s + (−0.994 − 0.106i)12-s + (−0.617 − 0.786i)13-s + (0.984 − 0.176i)14-s + (−0.271 − 0.962i)15-s + (−0.943 − 0.330i)16-s + (0.802 + 0.596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1063\)
Sign: $0.717 + 0.696i$
Analytic conductor: \(114.235\)
Root analytic conductor: \(114.235\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1063} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1063,\ (1:\ ),\ 0.717 + 0.696i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5287099671 + 0.2143428212i\)
\(L(\frac12)\) \(\approx\) \(0.5287099671 + 0.2143428212i\)
\(L(1)\) \(\approx\) \(0.5134161811 + 0.09326987869i\)
\(L(1)\) \(\approx\) \(0.5134161811 + 0.09326987869i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1063 \( 1 \)
good2 \( 1 + (-0.645 - 0.764i)T \)
3 \( 1 + (0.0620 + 0.998i)T \)
5 \( 1 + (-0.977 + 0.211i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.813 + 0.582i)T \)
13 \( 1 + (-0.617 - 0.786i)T \)
17 \( 1 + (0.802 + 0.596i)T \)
19 \( 1 + (0.00887 - 0.999i)T \)
23 \( 1 + (0.515 + 0.857i)T \)
29 \( 1 + (-0.999 + 0.0354i)T \)
31 \( 1 + (0.0620 + 0.998i)T \)
37 \( 1 + (-0.769 - 0.638i)T \)
41 \( 1 + (0.964 - 0.263i)T \)
43 \( 1 + (-0.861 - 0.507i)T \)
47 \( 1 + (0.959 - 0.280i)T \)
53 \( 1 + (0.999 - 0.0177i)T \)
59 \( 1 + (-0.734 + 0.678i)T \)
61 \( 1 + (-0.421 - 0.906i)T \)
67 \( 1 + (0.895 - 0.445i)T \)
71 \( 1 + (0.910 - 0.413i)T \)
73 \( 1 + (-0.996 + 0.0886i)T \)
79 \( 1 + (-0.758 - 0.651i)T \)
83 \( 1 + (0.842 - 0.537i)T \)
89 \( 1 + (0.949 - 0.314i)T \)
97 \( 1 + (0.710 - 0.703i)T \)
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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

−20.75858880968679578745365070243, −20.213502998190327412516844746338, −19.272428213995300673834326690555, −18.833537856181400719887867979505, −18.39821146942631778275652096344, −16.96442089404890319934687796470, −16.73627216431591930402978496824, −16.02218349350152137403862249551, −14.89348004654996462048588669181, −14.22250168005880155891539447948, −13.43092510786024725736780166789, −12.59334347081576021471370096528, −11.63206810303159504060599722297, −10.833635799574360587985691985981, −9.85769392014412846970798492020, −8.87552973487689355121502111056, −7.95543723185800286031743948542, −7.52679234665115002830624724844, −6.86292198222844239918017901095, −5.91782372237271132773172626707, −4.9305361198198295919485821234, −3.777342453670738870113002225031, −2.58029220104558459185687594068, −1.18516018958803764773393805668, −0.40669399057204464175513757454, 0.39485694367515493419390615734, 2.22895801582795153074200930881, 3.09598729823088091927000055757, 3.59172551520836468303231849554, 4.78784690206434709290261292445, 5.507347144011926771549973992977, 7.15191975127782348621396740406, 7.84827291089982478806352905651, 8.76783599182910883179344364435, 9.39869126024137936887669678731, 10.35097659071741149460042099790, 10.795569982859851184699537117762, 11.81872099983750509368105644146, 12.39071152890054566389051659960, 13.19816211158161281185986891836, 14.625925269511232929572917305464, 15.51946702839016049537097955882, 15.68470221662291174052164972207, 16.7741073920870987763610047711, 17.54693955246816921614814203957, 18.44065751732099478438902567309, 19.28633276031017449474161996741, 19.80939214583231879317729424670, 20.490072369216264874593210103436, 21.45916826796321970801854505853

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