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Properties

Label 1-1080-1080.709-r0-0-0
Degree $1$
Conductor $1080$
Sign $0.998 - 0.0581i$
Analytic cond. $5.01549$
Root an. cond. $5.01549$
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  + (−0.173 − 0.984i)7-s + (0.939 + 0.342i)11-s + (0.766 + 0.642i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.173 + 0.984i)23-s + (−0.766 + 0.642i)29-s + (0.173 − 0.984i)31-s + (−0.5 + 0.866i)37-s + (0.766 + 0.642i)41-s + (−0.939 − 0.342i)43-s + (−0.173 − 0.984i)47-s + (−0.939 + 0.342i)49-s + 53-s + (0.939 − 0.342i)59-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)7-s + (0.939 + 0.342i)11-s + (0.766 + 0.642i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.173 + 0.984i)23-s + (−0.766 + 0.642i)29-s + (0.173 − 0.984i)31-s + (−0.5 + 0.866i)37-s + (0.766 + 0.642i)41-s + (−0.939 − 0.342i)43-s + (−0.173 − 0.984i)47-s + (−0.939 + 0.342i)49-s + 53-s + (0.939 − 0.342i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.998 - 0.0581i$
Analytic conductor: \(5.01549\)
Root analytic conductor: \(5.01549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1080,\ (0:\ ),\ 0.998 - 0.0581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.658955825 - 0.04827068449i\)
\(L(\frac12)\) \(\approx\) \(1.658955825 - 0.04827068449i\)
\(L(1)\) \(\approx\) \(1.181706780 - 0.04095793459i\)
\(L(1)\) \(\approx\) \(1.181706780 - 0.04095793459i\)

Euler product

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

−21.41582714154600052694640972218, −20.849751186705900390659543420734, −19.69561206649437121281603208513, −19.27690646047642139695849478635, −18.28788856932088316543698267350, −17.73139439605656320437384271759, −16.720520762718019828467790594921, −15.98842404574722364820296402085, −15.19664472602721453780970987863, −14.515789450085571989793777387296, −13.583868199282300307374604677735, −12.687909610251427787083707153758, −12.03952496778300487408862929756, −11.17218173861575058878514145725, −10.365381017248514486232287967187, −9.25825268438718028534727432313, −8.72012307708595323733596054281, −7.884797608767451422771046938, −6.66123975460738701312565322558, −5.98000910399364917167856036997, −5.21312612051484228495953568720, −3.9618950580306812697672906088, −3.17030689678594806718646007548, −2.11270178046697340320494344517, −0.92905800046213936306166241963, 1.00600373320167197877095360989, 1.84498282101334488901521672539, 3.45481571458085422523617509783, 3.86219559906984394125278285853, 4.97564577304651997715676666125, 6.03464082194469591175442193449, 6.93961122475778833345449138571, 7.55107441270002130678809519607, 8.62057476827029254226585545161, 9.64147345948745941594870621452, 10.064688614338568118085516294713, 11.38719810314556942992161261207, 11.68673367680536428000940155809, 12.87738292788789719177750057093, 13.73377453662931265485559746488, 14.211248612550976109573101735025, 15.140049822513240870865233932175, 16.257786601122612318297399833877, 16.65958312228383096287082594555, 17.47686397324952810603404212045, 18.43354192512495302294910242750, 19.077995077008552484459431597958, 20.13316513764204119863225619112, 20.45618363323436180486930056407, 21.39698707393407532958756050728

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