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Properties

Label 1-1287-1287.578-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.755 + 0.655i$
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  + (−0.994 + 0.104i)2-s + (0.978 − 0.207i)4-s + (0.406 + 0.913i)5-s + (−0.951 − 0.309i)7-s + (−0.951 + 0.309i)8-s + (−0.5 − 0.866i)10-s + (0.978 + 0.207i)14-s + (0.913 − 0.406i)16-s + (−0.913 + 0.406i)17-s + (0.207 − 0.978i)19-s + (0.587 + 0.809i)20-s + 23-s + (−0.669 + 0.743i)25-s + (−0.994 − 0.104i)28-s + (0.669 + 0.743i)29-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)2-s + (0.978 − 0.207i)4-s + (0.406 + 0.913i)5-s + (−0.951 − 0.309i)7-s + (−0.951 + 0.309i)8-s + (−0.5 − 0.866i)10-s + (0.978 + 0.207i)14-s + (0.913 − 0.406i)16-s + (−0.913 + 0.406i)17-s + (0.207 − 0.978i)19-s + (0.587 + 0.809i)20-s + 23-s + (−0.669 + 0.743i)25-s + (−0.994 − 0.104i)28-s + (0.669 + 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.755 + 0.655i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (578, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.755 + 0.655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2234711775 + 0.5984534785i\)
\(L(\frac12)\) \(\approx\) \(0.2234711775 + 0.5984534785i\)
\(L(1)\) \(\approx\) \(0.6181532658 + 0.1365598533i\)
\(L(1)\) \(\approx\) \(0.6181532658 + 0.1365598533i\)

Euler product

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

−20.45259248447295851304533292528, −19.6758329669365927614698654875, −19.03880447136478423902471274522, −18.24255091894146515990870649781, −17.4317213009582902650584121695, −16.783127294028386815331734165324, −16.046956952782319354842324589363, −15.60884709498036942820016737783, −14.47359750654869841382962749932, −13.28437621293806068968452737178, −12.7144620085934175301683171568, −11.96416690241979623475646950844, −11.081963174923814131761009551439, −10.057672367170139445803052764398, −9.47272987624823330324249534062, −8.87465845892149637341127935560, −8.09376168971822862230057847834, −7.06018268772295617451285535242, −6.22960246785072657007775187366, −5.50430713829788212345905792032, −4.263157425763325639105624699979, −3.08189119073156805572685714558, −2.22249873751414134398731484466, −1.17120456624216309106662119700, −0.21978102814454036469150146018, 0.88183422532298591342577545917, 2.20156378246185597621029443795, 2.87091265172357684128745944098, 3.809999484103494820435224942418, 5.36013536638541469247007187894, 6.29096242760852841836331145500, 6.97217032540297769157859533379, 7.384110261387477377174856619065, 8.83226237957960658566576415594, 9.24096505194795694529132832540, 10.22585102589603067558669103361, 10.77588915402142270608170404902, 11.40423584817394285198153558723, 12.631307034315672405274277164450, 13.352351484139852809309804833293, 14.39252734825837375410578909380, 15.13319832783020337412215882861, 15.89209940870287849498049346571, 16.565506298411143815070765736409, 17.601904877933289429333857154557, 17.880241975016591836208731492026, 18.865348204060392849220788800023, 19.53008278148186839954398569024, 19.945893974100742187876752031, 21.094287417699465690964815738281

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