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Properties

Label 1-1287-1287.1181-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.593 - 0.804i$
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  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.743 − 0.669i)5-s + (−0.994 + 0.104i)7-s + (0.587 − 0.809i)8-s + (0.5 − 0.866i)10-s + (−0.913 + 0.406i)14-s + (0.309 − 0.951i)16-s + (−0.978 + 0.207i)17-s + (−0.994 − 0.104i)19-s + (0.207 − 0.978i)20-s + (−0.5 − 0.866i)23-s + (0.104 − 0.994i)25-s + (−0.743 + 0.669i)28-s + (0.809 − 0.587i)29-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.743 − 0.669i)5-s + (−0.994 + 0.104i)7-s + (0.587 − 0.809i)8-s + (0.5 − 0.866i)10-s + (−0.913 + 0.406i)14-s + (0.309 − 0.951i)16-s + (−0.978 + 0.207i)17-s + (−0.994 − 0.104i)19-s + (0.207 − 0.978i)20-s + (−0.5 − 0.866i)23-s + (0.104 − 0.994i)25-s + (−0.743 + 0.669i)28-s + (0.809 − 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.593 - 0.804i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.593 - 0.804i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.088910703 - 2.156866925i\)
\(L(\frac12)\) \(\approx\) \(1.088910703 - 2.156866925i\)
\(L(1)\) \(\approx\) \(1.527929534 - 0.8395156647i\)
\(L(1)\) \(\approx\) \(1.527929534 - 0.8395156647i\)

Euler product

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

−21.600680114638088644479954077971, −20.75952805319627010669792808908, −19.69392304019846383385889749335, −19.28799513796965423570138695788, −17.97024948046812631426270600765, −17.46266129601864540042951025653, −16.502480179642791156729732873165, −15.78005931599234132255562599261, −15.14274304367361902499842879010, −14.140073716019931412763378176855, −13.72417598597579571896742086772, −12.87411806426050691077436423835, −12.30152896356692095507432581282, −11.10606467826818725014321221347, −10.55872944597054455569032325129, −9.6106556335925174156481193025, −8.662241152209846679305830388785, −7.47376745263442425269907446202, −6.60887937051908797759105347107, −6.297325370861744740530198043494, −5.325229630986305272900544356274, −4.30815254755634994541914086977, −3.33744213251681126771830169940, −2.65688676342195948673337611822, −1.714134473832546278523539351233, 0.60664196535830108692420693921, 2.06359696949885492499037693310, 2.52821256725251952772644523247, 3.827314542237512315269620524164, 4.511729234914800426815054107413, 5.45651665178868700994451938375, 6.39210550050239516540177374322, 6.63172383605711929790805957940, 8.16780456271651999011152941381, 9.1105740229240490141902741192, 9.95238035648765619803890234036, 10.55663645012201710183479711150, 11.59472483513020180881334884353, 12.59916902030673579042660275229, 12.887354166024252293706266548789, 13.65685883347220451880387738527, 14.37350519895660280209154414055, 15.42975225368741012054505328631, 15.97030600080413017911523050469, 16.80273846219436706208526468727, 17.542777819258831736936744289066, 18.70107074486157984151953147348, 19.46656274688487349715134245589, 20.08110865525734460516480710952, 20.903226603055468324491969114224

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