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Properties

Label 1-1287-1287.38-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.871 + 0.490i$
Analytic cond. $138.307$
Root an. cond. $138.307$
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  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (0.913 − 0.406i)5-s + (0.978 − 0.207i)7-s + (0.309 + 0.951i)8-s + 10-s + (0.978 + 0.207i)14-s + (−0.104 + 0.994i)16-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.913 + 0.406i)20-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (0.809 + 0.587i)28-s + (0.978 − 0.207i)29-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (0.913 − 0.406i)5-s + (0.978 − 0.207i)7-s + (0.309 + 0.951i)8-s + 10-s + (0.978 + 0.207i)14-s + (−0.104 + 0.994i)16-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.913 + 0.406i)20-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (0.809 + 0.587i)28-s + (0.978 − 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.871 + 0.490i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.871 + 0.490i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.257413474 + 1.640327475i\)
\(L(\frac12)\) \(\approx\) \(6.257413474 + 1.640327475i\)
\(L(1)\) \(\approx\) \(2.551963685 + 0.5281834577i\)
\(L(1)\) \(\approx\) \(2.551963685 + 0.5281834577i\)

Euler product

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

−20.86085617456982996381257672580, −20.42267968541642903417773125033, −19.17370320560124602922527316790, −18.62106574103394706153015476014, −17.78299807024755148919984373025, −16.965309458384214301656068647224, −16.01935375788950784742145396694, −15.00509054426733136700999058813, −14.48388312522392750042659628499, −13.89203093733352485494092305892, −13.15148648142703477526417796760, −12.19152384348734082029507575163, −11.52819470415085768423127333814, −10.70649551754869217654950583976, −10.02463017338770307428591036428, −9.21184308630699686056079839433, −7.9491273121161229416157161119, −7.08400081738206243164746315381, −6.07294847824129850105500448605, −5.44842702984935737442727030615, −4.7357207135014942549226061012, −3.61080582496759818326862260611, −2.69143563870283490779544295826, −1.84099557002250060045768554673, −1.07245730727883172477064508421, 1.03208564291787398921359588521, 2.000712786799343637890705520727, 2.88578031008675413192371146752, 4.09058844417932376359257135483, 4.974281992058030169504972320802, 5.3826945084959027088852182728, 6.49362529277410367506701083338, 7.111252023067348422602489067670, 8.405247303209412416749154611952, 8.64239713825845657929430648069, 10.18724207755131705624036792357, 10.73285949876860368786518116288, 11.86635446806944741064489151810, 12.40852635751128039967133285018, 13.45880680316818003075760050598, 13.81133929671908840685655724936, 14.72284371961350725154847077319, 15.239808608517635526122501656255, 16.382311318120739338047508994069, 17.02872536158370419190625743235, 17.53242282895147909662276147966, 18.371250835007078310258830541138, 19.61535712695837464513163659587, 20.47193959705305690038725677754, 21.071469606306677969737170263063

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