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Properties

Label 1-1287-1287.1192-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.828 + 0.559i$
Analytic cond. $5.97680$
Root an. cond. $5.97680$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.978 + 0.207i)5-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (−0.5 − 0.866i)10-s + (−0.104 + 0.994i)14-s + (0.309 − 0.951i)16-s + (0.669 + 0.743i)17-s + (0.913 − 0.406i)19-s + (0.669 − 0.743i)20-s + (−0.5 + 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.978 + 0.207i)28-s + (−0.809 + 0.587i)29-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.978 + 0.207i)5-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (−0.5 − 0.866i)10-s + (−0.104 + 0.994i)14-s + (0.309 − 0.951i)16-s + (0.669 + 0.743i)17-s + (0.913 − 0.406i)19-s + (0.669 − 0.743i)20-s + (−0.5 + 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.978 + 0.207i)28-s + (−0.809 + 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.828 + 0.559i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1192, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.828 + 0.559i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4100611131 + 1.341030729i\)
\(L(\frac12)\) \(\approx\) \(0.4100611131 + 1.341030729i\)
\(L(1)\) \(\approx\) \(0.8211331830 + 0.6932931557i\)
\(L(1)\) \(\approx\) \(0.8211331830 + 0.6932931557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.978 + 0.207i)T \)
7 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−20.65612840460254128648289560089, −20.07895556020482729757867946529, −19.37703410932133743343218757077, −18.43618493162131766822556463752, −18.04077363402689603694851985937, −16.88698594245914395175450058801, −16.11579566454156202115236532425, −15.10070380139161044350426963560, −14.37884787324492114411108055507, −13.81767151926435837209061776917, −12.74929223834444461137165508770, −12.01211598977530711079658539920, −11.47439323698603685857154617596, −10.78742355754016643157889866438, −9.88752249553162975405858754536, −9.007603029535877800357418348840, −7.99447471867719398822945145659, −7.55076888121946604339212947767, −6.10269048783080661436332963553, −5.01087084667617762281263324557, −4.476623514184465764918916630799, −3.6183408209470271132245637399, −2.742775341402945176791618819062, −1.501953871007366774082941133477, −0.6179891390788420956833596808, 1.13176678820451860340451365022, 2.714608581486487702866475996283, 3.73260931554395193652582029821, 4.41023492183221395153921014296, 5.3818009092970531961313343337, 6.02976008639532110201823636708, 7.39309252230009755328241790271, 7.60713781213637532537228803133, 8.491389536993141818791225334964, 9.23830883584125215687810770851, 10.40048496614710890552531291629, 11.54598965366979369407586660057, 11.95412705805910747477241057489, 12.88948457240526142504415582048, 13.87857109848265169888451649021, 14.56009158943782615656307646998, 15.29953478955077387589787123368, 15.68292225958019894505719025911, 16.706180167549082221395070673529, 17.33759701745663348501259950949, 18.338281694564974710931653456070, 18.6878923266794672304770256041, 19.785588659216523878629191268223, 20.62715807984957861316257127129, 21.59004689727929302186581829446

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