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Properties

Label 1-1080-1080.517-r1-0-0
Degree $1$
Conductor $1080$
Sign $0.818 + 0.574i$
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  + (0.984 + 0.173i)7-s + (0.939 − 0.342i)11-s + (0.642 + 0.766i)13-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (0.984 − 0.173i)23-s + (0.766 + 0.642i)29-s + (0.173 + 0.984i)31-s + (−0.866 + 0.5i)37-s + (0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.984 + 0.173i)47-s + (0.939 + 0.342i)49-s − i·53-s + (−0.939 − 0.342i)59-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)7-s + (0.939 − 0.342i)11-s + (0.642 + 0.766i)13-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (0.984 − 0.173i)23-s + (0.766 + 0.642i)29-s + (0.173 + 0.984i)31-s + (−0.866 + 0.5i)37-s + (0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.984 + 0.173i)47-s + (0.939 + 0.342i)49-s − i·53-s + (−0.939 − 0.342i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.818 + 0.574i$
Analytic conductor: \(116.062\)
Root analytic conductor: \(116.062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1080,\ (1:\ ),\ 0.818 + 0.574i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.957206419 + 0.9338453217i\)
\(L(\frac12)\) \(\approx\) \(2.957206419 + 0.9338453217i\)
\(L(1)\) \(\approx\) \(1.428347663 + 0.1469532976i\)
\(L(1)\) \(\approx\) \(1.428347663 + 0.1469532976i\)

Euler product

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

−21.15304060473481627215664779755, −20.500740519452351277757993297530, −19.63242761503360281927513223018, −18.943476690085336657650945503374, −17.88117364851076283518771168275, −17.37578701112326829855769147308, −16.72135584416858012136340900425, −15.521349465190296274310489137988, −14.93233490220069724756180109140, −14.21830982255128245493504494747, −13.32575816717426316314032656257, −12.49844179691577167420508776135, −11.55331354787062814407710793059, −10.95427766366822250093734999408, −10.07013574902810234277467547196, −9.06621884816916295812974737940, −8.27601713274504302562080802391, −7.521544419574495882380881944279, −6.52917315577709426397629737053, −5.62161048054989099720847432785, −4.65112232645376164715094138627, −3.87457983466600634920270646567, −2.75379698908818369659405824026, −1.546241110993947354580729162255, −0.764355819317885129381366065708, 1.05800659830436664745128888076, 1.67646100856111585566466348218, 3.02517193775210682664115263804, 3.99225319589834056553529568192, 4.867713707817534510798470171221, 5.80151501185159997201132244534, 6.719204956572085190499944273581, 7.59187807202136479289505567059, 8.71787314661605702261046179063, 8.97662640214269004244776926309, 10.345731556523730802412474723619, 10.99622696405282185553354950223, 11.95282999211074375907374226765, 12.34919537857529922560286074533, 13.87129429466681108117511910603, 14.11398458837898619997858051559, 14.96018947377089326113329716682, 15.92393687026774485738201665705, 16.796439805981788950607813111355, 17.31105472149276941690572406290, 18.4167999303106886190535585958, 18.86130848385331907663995522574, 19.7585280055531403633454414620, 20.843164698705765451839838599586, 21.154171735570274826055373338114

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