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Properties

Label 1-1063-1063.1012-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  + (0.718 + 0.695i)2-s + (−0.579 + 0.814i)3-s + (0.0325 + 0.999i)4-s + (0.603 + 0.797i)5-s + (−0.983 + 0.182i)6-s + (0.766 + 0.642i)7-s + (−0.671 + 0.740i)8-s + (−0.328 − 0.944i)9-s + (−0.120 + 0.992i)10-s + (0.584 + 0.811i)11-s + (−0.833 − 0.552i)12-s + (0.710 − 0.703i)13-s + (0.103 + 0.994i)14-s + (−0.999 + 0.0295i)15-s + (−0.997 + 0.0650i)16-s + (−0.931 − 0.364i)17-s + ⋯
L(s)  = 1  + (0.718 + 0.695i)2-s + (−0.579 + 0.814i)3-s + (0.0325 + 0.999i)4-s + (0.603 + 0.797i)5-s + (−0.983 + 0.182i)6-s + (0.766 + 0.642i)7-s + (−0.671 + 0.740i)8-s + (−0.328 − 0.944i)9-s + (−0.120 + 0.992i)10-s + (0.584 + 0.811i)11-s + (−0.833 − 0.552i)12-s + (0.710 − 0.703i)13-s + (0.103 + 0.994i)14-s + (−0.999 + 0.0295i)15-s + (−0.997 + 0.0650i)16-s + (−0.931 − 0.364i)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: $\chi_{1063} (1012, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1063,\ (0:\ ),\ -0.839 - 0.543i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5157652127 + 1.744602932i\)
\(L(\frac12)\) \(\approx\) \(-0.5157652127 + 1.744602932i\)
\(L(1)\) \(\approx\) \(0.7209883835 + 1.183453468i\)
\(L(1)\) \(\approx\) \(0.7209883835 + 1.183453468i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1063 \( 1 \)
good2 \( 1 + (0.718 + 0.695i)T \)
3 \( 1 + (-0.579 + 0.814i)T \)
5 \( 1 + (0.603 + 0.797i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (0.584 + 0.811i)T \)
13 \( 1 + (0.710 - 0.703i)T \)
17 \( 1 + (-0.931 - 0.364i)T \)
19 \( 1 + (-0.990 - 0.135i)T \)
23 \( 1 + (-0.997 - 0.0768i)T \)
29 \( 1 + (-0.876 - 0.481i)T \)
31 \( 1 + (0.995 + 0.0945i)T \)
37 \( 1 + (-0.372 - 0.928i)T \)
41 \( 1 + (-0.589 + 0.807i)T \)
43 \( 1 + (-0.305 + 0.952i)T \)
47 \( 1 + (-0.350 + 0.936i)T \)
53 \( 1 + (0.963 - 0.268i)T \)
59 \( 1 + (0.421 + 0.906i)T \)
61 \( 1 + (-0.926 + 0.375i)T \)
67 \( 1 + (0.265 + 0.963i)T \)
71 \( 1 + (-0.697 - 0.716i)T \)
73 \( 1 + (0.742 + 0.669i)T \)
79 \( 1 + (0.915 + 0.402i)T \)
83 \( 1 + (-0.754 - 0.656i)T \)
89 \( 1 + (-0.943 - 0.330i)T \)
97 \( 1 + (0.830 + 0.557i)T \)
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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

−21.13437096146312313548988896216, −20.29329560491351200286761700667, −19.64164740677124337872602153542, −18.76161833786796031941955198351, −18.04552298449656436555887303098, −17.08358306059733809120785440075, −16.65913539491442148595350368727, −15.48087081597259129791296445334, −14.23246537695941979656276317404, −13.65741934766192836804673685841, −13.31410261792736862568238777201, −12.26506938211025609569696869777, −11.62455071637836247486769037402, −10.915623491614188997478439029237, −10.2025768207136083564203631937, −8.84916722983546199275360719442, −8.31145949268450750251480285110, −6.76059768014421494127070781623, −6.24213236402543580168410067103, −5.37520841183374142426460502520, −4.50387988490967828363422827993, −3.74778216155342847029020740199, −1.965163444374636462617777523657, −1.71830278785674138243628443797, −0.6049353634422900329546757538, 1.95270055727364738446320671287, 2.912289009663754522431361065664, 4.07381809910411096763809270220, 4.67902598525046722394304085740, 5.7431612252798325227579270820, 6.18758685899541202557166274674, 7.04549886772284797350391417365, 8.24388057379233581074407205288, 9.06644876808002729916487427026, 10.01711953782772049100979434891, 11.04610398635744948954927806407, 11.56226794829899993734204360348, 12.486154093948392518430671347046, 13.45555454900729118858876137673, 14.3912344659037429777883025338, 15.07935385644926349947410641252, 15.369410446140686004332597527928, 16.365655250880573809523100855315, 17.42433948019587665396028784910, 17.74040463724517482355445911717, 18.33601826586962811235286599563, 19.92290043932553102776661228068, 20.92041438863977075833526015298, 21.30702207715752900796459554265, 22.14991221433639385699466002637

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