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Properties

Label 1-1080-1080.893-r0-0-0
Degree $1$
Conductor $1080$
Sign $0.996 - 0.0880i$
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  + (−0.642 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.996 - 0.0880i$
Analytic conductor: \(5.01549\)
Root analytic conductor: \(5.01549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (893, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1080,\ (0:\ ),\ 0.996 - 0.0880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.335521819 - 0.05890342751i\)
\(L(\frac12)\) \(\approx\) \(1.335521819 - 0.05890342751i\)
\(L(1)\) \(\approx\) \(1.022382466 + 0.02542207518i\)
\(L(1)\) \(\approx\) \(1.022382466 + 0.02542207518i\)

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

−21.439979450541661567892637148097, −20.859150335539324438450632691322, −19.68623122637532205190183068148, −19.20972815107697280196841100507, −18.482471850516049706103973464973, −17.548009154793975324239332432654, −16.84889160389165694641532320307, −15.72633630809316534112712663834, −15.51190266497058593300218464717, −14.43139932774137347232702984205, −13.459182515947961273790417491750, −12.865082263139024738885127113434, −12.016825805316812550486038093047, −11.12149819838645512262243594589, −10.35034939852679192003112075530, −9.40530141766595868085746584401, −8.54731109219217727989324355987, −7.99032343829903496531589606425, −6.58713843307597486993220144167, −6.0395176024463734091540900652, −5.23619588516762400723458143760, −3.93230337046560714934798488578, −3.144766253021309553306001040628, −2.21970049885170540753899187923, −0.82235919637617964531920720047, 0.81353294805048774192076188194, 2.11341744286908685435482058112, 3.05646694068121057502360400039, 4.40027780325899538545107377824, 4.58744177554053439739863775808, 6.22168699958678960488312170253, 6.86025536228450277044099811989, 7.44032416881799940995124201110, 8.914441645169789099608843157087, 9.23237195720304203710664525955, 10.452321113498871438149664824020, 10.90764911189755038735417847903, 12.09878006077968601965575677693, 12.721791257175103883695856061759, 13.70347134541496107942659842466, 14.18035461665690920894272503226, 15.36359779173755447834866753204, 15.922577917770906690931332984474, 16.84435607411216741263441817762, 17.502148301471476638436042249710, 18.26913334012731720134949353157, 19.4222425226734162242024249067, 19.69384810027808765081461453257, 20.71136292936426133685551143558, 21.30608235062731194366555263521

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