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Properties

Label 1-1080-1080.653-r0-0-0
Degree $1$
Conductor $1080$
Sign $0.475 + 0.879i$
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  + (0.342 + 0.939i)7-s + (0.766 + 0.642i)11-s + (0.984 − 0.173i)13-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.342 + 0.939i)23-s + (−0.173 + 0.984i)29-s + (−0.939 − 0.342i)31-s + (0.866 − 0.5i)37-s + (−0.173 − 0.984i)41-s + (0.642 − 0.766i)43-s + (−0.342 − 0.939i)47-s + (−0.766 + 0.642i)49-s − i·53-s + (−0.766 + 0.642i)59-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)7-s + (0.766 + 0.642i)11-s + (0.984 − 0.173i)13-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.342 + 0.939i)23-s + (−0.173 + 0.984i)29-s + (−0.939 − 0.342i)31-s + (0.866 − 0.5i)37-s + (−0.173 − 0.984i)41-s + (0.642 − 0.766i)43-s + (−0.342 − 0.939i)47-s + (−0.766 + 0.642i)49-s − i·53-s + (−0.766 + 0.642i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.475 + 0.879i$
Analytic conductor: \(5.01549\)
Root analytic conductor: \(5.01549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1080,\ (0:\ ),\ 0.475 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.449228487 + 0.8641854264i\)
\(L(\frac12)\) \(\approx\) \(1.449228487 + 0.8641854264i\)
\(L(1)\) \(\approx\) \(1.179258857 + 0.2639396613i\)
\(L(1)\) \(\approx\) \(1.179258857 + 0.2639396613i\)

Euler product

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

−21.240538799453278817665181561027, −20.60756463573367651244155114429, −19.74277604254197954234635216785, −19.09896970639790162539200709726, −18.197906136020282224915681165774, −17.36493733693176251914200691225, −16.61842035683262383451214350305, −16.108607107134958580588580851200, −14.832193659901289374949327875766, −14.329002641732882720981415349938, −13.45509669134890565467492206436, −12.81215051978761885939316480158, −11.580619424244011815749977163078, −11.07263388556515941489053546663, −10.25894662432512567844187391645, −9.274769618746329436026580574869, −8.37394001383963926460430007583, −7.695239357595996399228599616165, −6.53016411388948052322168007970, −6.03956348532567783488682084888, −4.67480830210287611888737190928, −3.97170664084819180785021105869, −3.11034020015598842667236069910, −1.688063341193154265364034842744, −0.78873644430568802413232932270, 1.32036937076722595740739370028, 2.10895022067413119723333608102, 3.38370105775918791947056984419, 4.14698621561161817673692377973, 5.443214784275668334107774945591, 5.875253532089378080027757589332, 7.05261083885075389814583474585, 7.90404752848858169676136409178, 8.84842478116611037805926025042, 9.43854010730060959973334006549, 10.46714302083492950937644593141, 11.38803136407919631007563539312, 12.12717832779370925892025098559, 12.7313551630494864783779460860, 13.867112520681815285019294735338, 14.59906584668549618173336903364, 15.259329037633943663085016621599, 16.13257092254129157452482978797, 16.88420404760876164816213780294, 17.8803514906332275426003850040, 18.43204526373321668866917010275, 19.14563412304213351519153643838, 20.16848082765155889060442045892, 20.79180439628462970988424101315, 21.60664697435382285964222893925

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