Properties

Label 1-288-288.85-r0-0-0
Degree $1$
Conductor $288$
Sign $-0.915 + 0.402i$
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  + (−0.258 + 0.965i)5-s + (−0.866 − 0.5i)7-s + (−0.965 + 0.258i)11-s + (0.965 + 0.258i)13-s − 17-s + (−0.707 + 0.707i)19-s + (−0.866 + 0.5i)23-s + (−0.866 − 0.5i)25-s + (0.258 + 0.965i)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 + (−0.965 + 0.258i)43-s + (0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)5-s + (−0.866 − 0.5i)7-s + (−0.965 + 0.258i)11-s + (0.965 + 0.258i)13-s − 17-s + (−0.707 + 0.707i)19-s + (−0.866 + 0.5i)23-s + (−0.866 − 0.5i)25-s + (0.258 + 0.965i)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 + (−0.965 + 0.258i)43-s + (0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.915 + 0.402i$
Analytic conductor: \(1.33746\)
Root analytic conductor: \(1.33746\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 288,\ (0:\ ),\ -0.915 + 0.402i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08088323918 + 0.3846502041i\)
\(L(\frac12)\) \(\approx\) \(0.08088323918 + 0.3846502041i\)
\(L(1)\) \(\approx\) \(0.6621977976 + 0.1860180479i\)
\(L(1)\) \(\approx\) \(0.6621977976 + 0.1860180479i\)

Euler product

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

−25.19060054547239904023191827415, −24.11400828504215975465957823696, −23.49173763466290163189503086773, −22.42634933397982697876414539204, −21.46099621797385419318850589899, −20.5610851449890743804678882005, −19.74567135446955396860761289289, −18.806497536152664926662787341450, −17.862697248486474339692778642854, −16.73962501499763956474601337130, −15.667056976061951811767642989733, −15.55713179634097898979218257666, −13.648467188475128142401010393133, −13.05436866210663672970182172701, −12.22406735234262681812763321622, −11.05803684905375311517398646893, −9.977941604617838001571457138413, −8.746066856975801702352174297336, −8.3034267637905991042121568832, −6.72651274036696548259536282242, −5.72444043132207585707546750274, −4.64647006816884468466849143687, −3.42203589652227666355669315341, −2.09587483055855460931716455207, −0.24156471344193900138957343156, 2.050604412369823462737936019500, 3.31469658961300742367419999730, 4.16699123702763432289926683162, 5.85105227351822568890020534711, 6.72864978063085076633604929570, 7.63055866413066313777871894681, 8.835519370947514860585173456581, 10.22722891158229067687037567256, 10.65946145215403201461489930373, 11.83598328668751465128585339815, 13.10736555007633377953069301615, 13.736166353222406236085099285705, 14.97522160347979838482783495552, 15.77247431119512222685231563495, 16.59524511949530253406410588099, 17.977532693291686612216110677136, 18.54074013897498950858014822560, 19.54327996442716854458301571439, 20.36571345282668018423233993257, 21.51886639640601863036033190979, 22.39894120484047524902681974854, 23.285256668763286172684658906301, 23.733855923869870261617541044255, 25.30314550610236816899326803476, 26.06008795023056662162477317480

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