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Properties

Label 1-1287-1287.1150-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.655 - 0.755i$
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  + (0.994 − 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

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

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.655 - 0.755i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1150, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.655 - 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8279565101 - 1.814799973i\)
\(L(\frac12)\) \(\approx\) \(0.8279565101 - 1.814799973i\)
\(L(1)\) \(\approx\) \(1.405296977 - 0.6664503405i\)
\(L(1)\) \(\approx\) \(1.405296977 - 0.6664503405i\)

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

−21.61354338825189184303054880349, −20.50673832296472449552349870662, −19.93392435716577996009050298797, −18.896487653048882469291890667938, −18.63826504466054012015369595133, −17.2873788406911474457116608813, −16.3211743197556269652984757432, −15.91359029462746170813313114839, −14.95514592582160640979448971993, −14.46935208552327163723702522637, −13.67456325064497563550438210995, −12.7042547035518477328800093242, −12.14029734076177961355377354750, −11.402613307709651288139139062455, −10.374417912964280553847259402780, −9.91980106356067781449990128656, −8.47169988894384658793448855654, −7.528342158336039944092100396117, −6.89776611020023631554819324030, −5.97479849863202108520873711813, −5.49003027568846686376718056536, −3.98304345231986323786944702760, −3.53782622434344459839644634621, −2.73662263237248598787603447583, −1.68423888208043380237805794030, 0.51652681543450990412631632234, 1.75489316822815951543279169971, 2.95191636708324959776813143790, 3.756988887071540665025132216823, 4.47337921651887594980745882358, 5.44793665337854470330730944626, 6.095041671841204574271146092502, 7.24397482101841351330141954796, 7.74872295549854492867554342466, 9.08784561323642518202209081078, 9.777802350103876084362089570, 10.77048154925347934282225853646, 11.674704487836811897273982822647, 12.36324088988183028847912930187, 12.96870403532343500562553759649, 13.65393338446910292811338506989, 14.434319810208082002999017110796, 15.53515270339029770546518558579, 16.00198795429327836106544445285, 16.6086211582238739917744367103, 17.41138837857050094751814094391, 18.79021612029016753353010179356, 19.45757675883766310474467179768, 20.19323526332676477990847575204, 20.600039453478274306473045977293

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