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Properties

Label 1-1080-1080.997-r1-0-0
Degree $1$
Conductor $1080$
Sign $-0.851 + 0.524i$
Analytic cond. $116.062$
Root an. cond. $116.062$
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  + (0.642 + 0.766i)7-s + (−0.173 − 0.984i)11-s + (−0.342 − 0.939i)13-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (0.642 − 0.766i)23-s + (−0.939 − 0.342i)29-s + (0.766 + 0.642i)31-s + (0.866 + 0.5i)37-s + (−0.939 + 0.342i)41-s + (−0.984 + 0.173i)43-s + (0.642 + 0.766i)47-s + (−0.173 + 0.984i)49-s − i·53-s + (0.173 − 0.984i)59-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)7-s + (−0.173 − 0.984i)11-s + (−0.342 − 0.939i)13-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (0.642 − 0.766i)23-s + (−0.939 − 0.342i)29-s + (0.766 + 0.642i)31-s + (0.866 + 0.5i)37-s + (−0.939 + 0.342i)41-s + (−0.984 + 0.173i)43-s + (0.642 + 0.766i)47-s + (−0.173 + 0.984i)49-s − i·53-s + (0.173 − 0.984i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.851 + 0.524i$
Analytic conductor: \(116.062\)
Root analytic conductor: \(116.062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (997, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1080,\ (1:\ ),\ -0.851 + 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05669063417 + 0.2002346777i\)
\(L(\frac12)\) \(\approx\) \(0.05669063417 + 0.2002346777i\)
\(L(1)\) \(\approx\) \(0.9205281299 - 0.04287529192i\)
\(L(1)\) \(\approx\) \(0.9205281299 - 0.04287529192i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.642 + 0.766i)T \)
11 \( 1 + (-0.173 - 0.984i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (0.642 + 0.766i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.342 + 0.939i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (0.342 - 0.939i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.984 + 0.173i)T \)
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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

−20.90949596966904611237545458198, −20.21549248563888998319029103273, −19.484383649176150410166345303538, −18.589379494452674393068188420037, −17.75019064351310837656337392250, −17.02701443095383175168622927301, −16.521485953915362552747519814494, −15.1167155116066611574115712495, −14.9138885875381723886471788168, −13.79280079480038513718008934944, −13.17671132014768879262558523360, −12.18108232627134720292428324760, −11.39396695491260774284256937738, −10.588436608215571192488767610366, −9.79955346091000021647375070972, −8.91121476884438810449694598978, −7.9002285511233054149660914756, −7.193988749791190403425489172129, −6.41803352273784400452462455237, −5.16264091374149886055076446209, −4.39264045966967003514634498652, −3.69053861941387422170379374089, −2.13620335348811463290223834108, −1.57636897369494983208293082722, −0.04316010816178303311265164234, 1.079419561149190679382351778624, 2.47724332117577106460665340828, 2.98720011785874640394585526044, 4.45290315388510331399654702327, 5.15663914656578841733010594754, 6.02682279869470123538871685166, 6.9563019933178907223089229659, 8.0971929402015577225006272263, 8.61707937047877067802499539548, 9.46170599367520919320709151519, 10.6489281186173484628877775399, 11.21276001915050258959819399162, 12.01488766194536014529366507931, 13.02109907681317622797482082536, 13.59748485595050496474107686725, 14.73065263128199144277159427267, 15.251668971783185571493083088699, 16.012736089055068807998265473930, 17.040027609917398126763906485014, 17.70757965279637747881396962015, 18.525014683752723769778398323820, 19.11687330768215643554108251988, 20.15926935694980554189486298673, 20.74682099920882758668905215545, 21.86011621753328830604869258998

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