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Properties

Label 1-1063-1063.719-r0-0-0
Degree $1$
Conductor $1063$
Sign $-0.999 + 0.0265i$
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  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + 5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + 12-s + 13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + 5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + 12-s + 13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1063\)
Sign: $-0.999 + 0.0265i$
Analytic conductor: \(4.93655\)
Root analytic conductor: \(4.93655\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1063} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1063,\ (0:\ ),\ -0.999 + 0.0265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01223521266 - 0.9202190714i\)
\(L(\frac12)\) \(\approx\) \(0.01223521266 - 0.9202190714i\)
\(L(1)\) \(\approx\) \(0.5148503463 - 0.5730686572i\)
\(L(1)\) \(\approx\) \(0.5148503463 - 0.5730686572i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1063 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−21.87481702384825769078369618787, −21.3280927356483441588302970144, −20.43237125182120031206470542406, −19.42744844742703594077705263216, −18.238102855696102343699560112261, −18.01741630323694172898406189927, −17.21276445303203511166406344321, −16.24306748222245988445584149445, −15.90659614916475141758307899401, −14.96248933266377197461511592848, −14.43556345380557595311892217304, −13.26455091379723489870943019836, −12.57240511793109361877070638665, −11.25212807802658718128470160363, −10.43461311658463840711868805403, −9.70294376114640850396782624581, −9.218272909341694686930573992104, −8.43628956992403211014845747100, −7.10629396413693172406428596985, −6.239476191921194174512840262539, −5.54974252832777156881097967422, −5.104846445119632976552541430665, −3.84994392643045987141198540968, −2.54048331029566771205266644991, −1.26529550048495101110408487135, 0.55615315327092306107304935081, 1.379947515424088261156904201474, 2.36847263575880451503833205114, 3.320072796926147290747870893679, 4.40809457439868200694589498539, 5.90390024173122798290051055135, 6.1630932840526864860895770034, 7.584967870628490398173637531398, 8.083324110991575269176594115603, 9.192838585594474185796975006012, 10.13329377346563030645720426904, 10.74451053769413274362347593846, 11.355167781776051760131093743598, 12.65325946485024607971678947161, 12.93345099418147311616426631485, 13.78591849112385565829201802406, 14.24023230163962250524553391198, 16.337937416904569658534234696333, 16.56471346906122045994025254300, 17.3484243724604253219366137483, 18.20153660365390574194496089494, 18.754498700795538412681567990099, 19.26336665198539454520239150659, 20.49231839990058864881714000584, 20.87497688611315087259366938900

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