Properties

Label 1-1287-1287.16-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.967 + 0.254i$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.913 − 0.406i)5-s + (0.669 + 0.743i)7-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)10-s + (−0.978 − 0.207i)14-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (0.669 − 0.743i)19-s + (−0.104 − 0.994i)20-s + (−0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (0.913 − 0.406i)28-s + (0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.913 − 0.406i)5-s + (0.669 + 0.743i)7-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)10-s + (−0.978 − 0.207i)14-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (0.669 − 0.743i)19-s + (−0.104 − 0.994i)20-s + (−0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (0.913 − 0.406i)28-s + (0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.967 + 0.254i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.967 + 0.254i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.387774487 + 0.1795697845i\)
\(L(\frac12)\) \(\approx\) \(1.387774487 + 0.1795697845i\)
\(L(1)\) \(\approx\) \(0.9636680432 + 0.1606962134i\)
\(L(1)\) \(\approx\) \(0.9636680432 + 0.1606962134i\)

Euler product

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

−20.83102585952009230128270609998, −20.28106654751266614429341526881, −19.50749478293486210105471947988, −18.48307576619660718228225546656, −17.893679603295304363922571114301, −17.54499802683156512909183385631, −16.51495363741733827603373570761, −15.99934739075855810606861412417, −14.628999911338588570103674853408, −13.960244310352134655100695657493, −13.30165987164926027321731515297, −12.28035208617426890682826587444, −11.41831893874220797020042552971, −10.7559273842738328158315378632, −10.02242588858671740468006869943, −9.416205123709079600354789674680, −8.49056119030752055596205454596, −7.45728959822725176969789509691, −7.03905339469640949948882142382, −5.827450916256788203637471840339, −4.80167781038897218191367617770, −3.66090917283350125794272413496, −2.8111092381602763501969141119, −1.77552120898064606862264459395, −1.06054118037494739386928684695, 0.88104538552953357129901867973, 1.94660492820095424727652284823, 2.54795912442686071783981624458, 4.39174640393734346491702651328, 5.21937910942893418315469883051, 5.97801036519884478698128453023, 6.572765610528081585030302692816, 7.8801793342230780124200179994, 8.3849913846672020508075579940, 9.250297127294995877937852565661, 9.819657007443901319503096805377, 10.7529873688530370770142605366, 11.540400277665903071596325411343, 12.51913775443093972317575228032, 13.52438237328534620997575600502, 14.27265776897176173447459037438, 15.04666208398360513398211295378, 15.69751323340918740943961886056, 16.67065918727654172579762509439, 17.2800117560218926799789165056, 17.93044821861807327792794609648, 18.49948357634745738763729899649, 19.34383663063256766538745987725, 20.3071216494677409544662540938, 20.86108113158722761775786366115

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