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Properties

Label 1-1287-1287.7-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.995 - 0.0957i$
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  + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.743 + 0.669i)5-s + (−0.587 − 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.5 − 0.866i)10-s + (0.913 − 0.406i)14-s + (0.669 + 0.743i)16-s + (0.669 + 0.743i)17-s + (−0.406 − 0.913i)19-s + (0.951 − 0.309i)20-s − 23-s + (0.104 − 0.994i)25-s + (0.207 + 0.978i)28-s + (0.104 + 0.994i)29-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.743 + 0.669i)5-s + (−0.587 − 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.5 − 0.866i)10-s + (0.913 − 0.406i)14-s + (0.669 + 0.743i)16-s + (0.669 + 0.743i)17-s + (−0.406 − 0.913i)19-s + (0.951 − 0.309i)20-s − 23-s + (0.104 − 0.994i)25-s + (0.207 + 0.978i)28-s + (0.104 + 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.995 - 0.0957i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.995 - 0.0957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01882734988 + 0.3924146791i\)
\(L(\frac12)\) \(\approx\) \(0.01882734988 + 0.3924146791i\)
\(L(1)\) \(\approx\) \(0.5471531215 + 0.3074341859i\)
\(L(1)\) \(\approx\) \(0.5471531215 + 0.3074341859i\)

Euler product

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

−20.59435262948861012342274168651, −19.853178428634093135596207524302, −19.05112369192182202441801288092, −18.71200503475399571873275447990, −17.745632804212190055111671917498, −16.78267460618603456222885444254, −16.15063647287494683108973570770, −15.354986297109093896173346104585, −14.292549104465813323008813914081, −13.44920795757725556950867719118, −12.51825611968060687229134925782, −12.08210778603039379427470883803, −11.57048469131848379737107359671, −10.35867910701225522466801132778, −9.714419320634530635999057257179, −8.8517390312446349727726794075, −8.24582766746738087344158891219, −7.42452364245019238828546872684, −6.00557828884514005048989538782, −5.159656826562113149569886065982, −4.20631486181426324569911423230, −3.43960166577789896861656874812, −2.53326142668575822290796110984, −1.45393916056977959242123300503, −0.20577846790409098506655042469, 1.044670976333429792759296102619, 2.73824978400580016081334684317, 3.92030139792129670373863908382, 4.26430655474082720372795996651, 5.653251135819345466649721240254, 6.43491315662877379459273964331, 7.190232636913041867579003114273, 7.756290992668977866337622036002, 8.61307934582060958544049652088, 9.64012157803779385601082448499, 10.39633741292392124448840981428, 11.032305771168991628429135257956, 12.2776619669843480051521997983, 13.074887233696017596610293520586, 13.94052074642441217210759294544, 14.61022214319264437279814225158, 15.33365476923351435243720195162, 16.07416635087810207425789194630, 16.686938236152674316768500036381, 17.50934607898686554928879476499, 18.274164831440561329963146571012, 19.1464976570977250772288004018, 19.56462979008365178347697365969, 20.39755250338747128169149912785, 21.8299458662423681337494159791

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