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Properties

Label 1-1080-1080.43-r0-0-0
Degree $1$
Conductor $1080$
Sign $0.818 - 0.574i$
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

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

Invariants

Degree: \(1\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.818 - 0.574i$
Analytic conductor: \(5.01549\)
Root analytic conductor: \(5.01549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1080,\ (0:\ ),\ 0.818 - 0.574i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.573230715 - 0.4968047324i\)
\(L(\frac12)\) \(\approx\) \(1.573230715 - 0.4968047324i\)
\(L(1)\) \(\approx\) \(1.170457419 - 0.1600917042i\)
\(L(1)\) \(\approx\) \(1.170457419 - 0.1600917042i\)

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

−21.531857949527237087827054763941, −20.81667204128667823529763352883, −19.96901456875295749557695737254, −19.13653236587641831774394634124, −18.532089684257950672405950445320, −17.5014024043267220708875294602, −17.09489636910951725713414478251, −15.9539205585199698867104668245, −15.23938668801586531937182922022, −14.512962630144254207503789364189, −13.86213154586347973965047353283, −12.60426225309322866722795667685, −12.029932428387581337147583399638, −11.48246423335509837495432480566, −10.23505628300869111477114528235, −9.47818579150588492332378299041, −8.811743853877223249922317919247, −7.76966442433489815175932101299, −6.96819707866291798037908248342, −6.0405989449395816063973298534, −4.98436839698836176291849652050, −4.44580349494642334646354491899, −2.976075319145381461421320000382, −2.313849923512669324584794565966, −1.07282451413400231185070686985, 0.863269277447790596593891016683, 1.78444370315731999212705809591, 3.262439909615172210087514712606, 3.85271359042904189023365713050, 4.98445164351174664794416781611, 5.7924149661590372354523132741, 6.9090217103635725226537136977, 7.60561932235738460067965485600, 8.40074448821462353247935613440, 9.50396365607155945485380221274, 10.205608326800090892625929233980, 11.016630643937030183719238373281, 11.92003913558798786664189012339, 12.592916491616164115636967182963, 13.76589036240049833542790518301, 14.223566418361146042700064707057, 14.95430737068082211334414043761, 16.0679081794782704897854040237, 16.8848991809336139927354619494, 17.27713321565260923822052140223, 18.22507136206616457235301927419, 19.43464449996006346890847291989, 19.53007372767465173813195475237, 20.72273711306985431120299993399, 21.250299506894477227039553810281

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