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Properties

Label 1-288-288.13-r0-0-0
Degree $1$
Conductor $288$
Sign $0.691 - 0.722i$
Analytic cond. $1.33746$
Root an. cond. $1.33746$
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  + (−0.965 − 0.258i)5-s + (0.866 + 0.5i)7-s + (−0.258 − 0.965i)11-s + (0.258 − 0.965i)13-s − 17-s + (0.707 + 0.707i)19-s + (0.866 − 0.5i)23-s + (0.866 + 0.5i)25-s + (0.965 − 0.258i)29-s + (−0.5 − 0.866i)31-s + (−0.707 − 0.707i)35-s + (0.707 − 0.707i)37-s + (0.866 − 0.5i)41-s + (−0.258 − 0.965i)43-s + (0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)5-s + (0.866 + 0.5i)7-s + (−0.258 − 0.965i)11-s + (0.258 − 0.965i)13-s − 17-s + (0.707 + 0.707i)19-s + (0.866 − 0.5i)23-s + (0.866 + 0.5i)25-s + (0.965 − 0.258i)29-s + (−0.5 − 0.866i)31-s + (−0.707 − 0.707i)35-s + (0.707 − 0.707i)37-s + (0.866 − 0.5i)41-s + (−0.258 − 0.965i)43-s + (0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.691 - 0.722i$
Analytic conductor: \(1.33746\)
Root analytic conductor: \(1.33746\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 288,\ (0:\ ),\ 0.691 - 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.001223384 - 0.4275744065i\)
\(L(\frac12)\) \(\approx\) \(1.001223384 - 0.4275744065i\)
\(L(1)\) \(\approx\) \(0.9764425371 - 0.1543031163i\)
\(L(1)\) \(\approx\) \(0.9764425371 - 0.1543031163i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 + (0.258 - 0.965i)T \)
17 \( 1 - T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.965 - 0.258i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (-0.258 - 0.965i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.965 - 0.258i)T \)
61 \( 1 + (0.965 - 0.258i)T \)
67 \( 1 + (-0.258 + 0.965i)T \)
71 \( 1 - iT \)
73 \( 1 + iT \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.965 + 0.258i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−25.77275976366205090296322139718, −24.54464212092279528741804127194, −23.677667410850212171840699284741, −23.19646025808294664763960182969, −22.06517541184511379354647490334, −21.03482646618352016370900688393, −20.095991263171698694424101040450, −19.46425116085874326474214432303, −18.205603367357726668775005404940, −17.598869811212290196084608365661, −16.349643516440334270409150777470, −15.46858176477612574157552812029, −14.64793679332257389838114122806, −13.6822053343769643490783688861, −12.50079826754508061307646146409, −11.372291075402059854714651834478, −10.96054280469099054271736211289, −9.527878131829776684138372479077, −8.41263795115000516419299046439, −7.37891007726843947477488319670, −6.73619412679970275177731456880, −4.82973036555858981501337514994, −4.32991072158884635638527655952, −2.88230623035627780556351846400, −1.37980648346346141628522781720, 0.84108233419696394975111337, 2.58810531639685826836266660445, 3.80051820937099124307557168207, 4.9620417136847753582955084409, 5.92334557582298565867292932722, 7.46934936115208163607569320615, 8.26809422293401034766737929280, 8.97974320709461683551286782408, 10.66625306487692675559428075047, 11.304321662161963227201321356784, 12.2555964935551899923372955709, 13.25658436099893517213261754096, 14.43519643208320974267232276934, 15.398291090988554279159148688959, 16.02120233340627029658891391440, 17.17086960644147196450647856049, 18.25829862577519324844243506859, 18.9399111886107257213281223174, 20.09232272050761735970836176767, 20.74406945620157787823232593425, 21.80444479445648105182394048939, 22.763475638813066807585163856818, 23.68763169289123904474656720778, 24.52336428735593849725872727206, 25.08850456563937097802013681228

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