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Properties

Label 1-1287-1287.785-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.371 - 0.928i$
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  + (0.743 + 0.669i)2-s + (0.104 + 0.994i)4-s + (−0.743 + 0.669i)5-s + (−0.406 + 0.913i)7-s + (−0.587 + 0.809i)8-s − 10-s + (−0.913 + 0.406i)14-s + (−0.978 + 0.207i)16-s + (0.309 − 0.951i)17-s + (−0.587 + 0.809i)19-s + (−0.743 − 0.669i)20-s + (−0.5 + 0.866i)23-s + (0.104 − 0.994i)25-s + (−0.951 − 0.309i)28-s + (−0.913 − 0.406i)29-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.104 + 0.994i)4-s + (−0.743 + 0.669i)5-s + (−0.406 + 0.913i)7-s + (−0.587 + 0.809i)8-s − 10-s + (−0.913 + 0.406i)14-s + (−0.978 + 0.207i)16-s + (0.309 − 0.951i)17-s + (−0.587 + 0.809i)19-s + (−0.743 − 0.669i)20-s + (−0.5 + 0.866i)23-s + (0.104 − 0.994i)25-s + (−0.951 − 0.309i)28-s + (−0.913 − 0.406i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.371 - 0.928i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (785, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.371 - 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4362189031 + 0.6442056882i\)
\(L(\frac12)\) \(\approx\) \(-0.4362189031 + 0.6442056882i\)
\(L(1)\) \(\approx\) \(0.7274444790 + 0.7490324742i\)
\(L(1)\) \(\approx\) \(0.7274444790 + 0.7490324742i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.743 + 0.669i)T \)
5 \( 1 + (-0.743 + 0.669i)T \)
7 \( 1 + (-0.406 + 0.913i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.587 + 0.809i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.913 - 0.406i)T \)
31 \( 1 + (-0.207 + 0.978i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (0.406 + 0.913i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.994 + 0.104i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.994 + 0.104i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.951 + 0.309i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.207 - 0.978i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.743 - 0.669i)T \)
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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

−20.35054162345065999602803868853, −19.997775244147010520417846134750, −19.15970128825582602413174233418, −18.62498966672996070863505136116, −17.1770367863933651212296635292, −16.71745697044295255486351879258, −15.63459558933171325275425091679, −15.15951374753831715481020224341, −14.12198376981137300769175859867, −13.42323149119044421848635510279, −12.58666332971286611384328206789, −12.25473007776237285988438112194, −10.99819355584937738754869066368, −10.698727821924717618375002264836, −9.6210632458218052161460066552, −8.80361256942293517877579237295, −7.74143815489282409897756225150, −6.83856305102408628315143663347, −5.92341351862945955249337419566, −4.914024316008283498426462337882, −4.04353935207546383681580541278, −3.69262773346418609223018750577, −2.42765307711264684731184225830, −1.27934229365720189602413528206, −0.235596165385055802356158793263, 2.07589816548843065629047680032, 3.0537878921210674065108355361, 3.65240569692657358013013764467, 4.662657416421835683295078302909, 5.685118282713298422258368197, 6.2799412557972526882232970603, 7.27842561026784846116950106375, 7.83565455647729118372831623336, 8.779107439964028679068732752012, 9.66325068323991463446851400941, 10.898455730492089441942992450024, 11.71495504711869371672106999496, 12.267413221538918212229950487140, 13.02194247691351823739413906413, 14.12001480367848623559712624184, 14.61473365954358683567142131114, 15.47398548212711060831569344543, 15.93425868914940720032548045039, 16.63075895131460721364068609816, 17.77199036715566317346594818227, 18.41931031202405509652344731327, 19.181870768771188227623804636211, 20.03307130472004414650252114677, 21.070503673443559053184544067098, 21.67032329640896698770316351352

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