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Properties

Label 1-1080-1080.907-r0-0-0
Degree $1$
Conductor $1080$
Sign $0.879 + 0.475i$
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

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

Invariants

Degree: \(1\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(5.01549\)
Root analytic conductor: \(5.01549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (907, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1080,\ (0:\ ),\ 0.879 + 0.475i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.503593766 + 0.3802463079i\)
\(L(\frac12)\) \(\approx\) \(1.503593766 + 0.3802463079i\)
\(L(1)\) \(\approx\) \(1.124303844 + 0.1162635054i\)
\(L(1)\) \(\approx\) \(1.124303844 + 0.1162635054i\)

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

−21.43021948175818043136784968508, −20.45828354987449384556600832007, −19.79893088436859602507283777498, −19.38358038271616526704524056529, −18.02826084433395281273409528094, −17.6064710545107089719032853704, −16.77894527375850332303116807127, −15.88980798804761889844896056021, −15.24901851655727409292523042840, −14.24900874505920381986916311665, −13.38410961376588430403010745084, −13.0196466879810915359908748412, −11.68141963685927070960138269656, −11.19962676853095905171296619130, −10.11823209830207598688221037104, −9.524798426828071669238516475478, −8.51987316840004137011327138468, −7.60889230428006417007539941062, −6.69107883250133731730627810803, −6.13492399320658117964328244481, −4.75299542086412772669412494898, −4.037362066603952838297764094784, −3.196534453541806753268909429222, −1.866984319634306799585976220457, −0.83254232717364150986656404299, 1.014909686955408683241223343488, 2.21077361348447948421171801535, 3.199864225854351036227273964137, 4.075703925367348687656042907253, 5.19835159116490736983976524203, 6.20992021341343553203129785125, 6.583176496540167854988484544501, 7.99238182749015705479849186765, 8.767862464988981820060628410417, 9.31218499598366150325821099460, 10.39226632212035449017786295229, 11.33609110684526723613840233375, 11.96144343498835169360681445992, 12.790776050820538328115758611226, 13.75699214568794168777365887894, 14.35985356241903955998512353652, 15.40635826532230143768888977330, 16.05796059112702892403165926155, 16.67706863974384478245064365003, 17.77010632654778056530922989697, 18.58577115279303403534769914686, 18.954303827016461092305172420983, 20.05896134943539258207140360761, 20.6772920553685492664454156664, 21.65891910116135661352581041781

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