Properties

Label 1-72e2-5184.11-r0-0-0
Degree $1$
Conductor $5184$
Sign $-0.988 + 0.153i$
Analytic cond. $24.0743$
Root an. cond. $24.0743$
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  + (0.585 − 0.810i)5-s + (−0.989 + 0.144i)7-s + (0.355 + 0.934i)11-s + (−0.756 − 0.653i)13-s + (−0.342 − 0.939i)17-s + (0.0436 + 0.999i)19-s + (−0.989 − 0.144i)23-s + (−0.314 − 0.949i)25-s + (−0.561 + 0.827i)29-s + (0.396 − 0.918i)31-s + (−0.461 + 0.887i)35-s + (0.887 − 0.461i)37-s + (0.664 + 0.747i)41-s + (0.631 − 0.775i)43-s + (0.918 − 0.396i)47-s + ⋯
L(s)  = 1  + (0.585 − 0.810i)5-s + (−0.989 + 0.144i)7-s + (0.355 + 0.934i)11-s + (−0.756 − 0.653i)13-s + (−0.342 − 0.939i)17-s + (0.0436 + 0.999i)19-s + (−0.989 − 0.144i)23-s + (−0.314 − 0.949i)25-s + (−0.561 + 0.827i)29-s + (0.396 − 0.918i)31-s + (−0.461 + 0.887i)35-s + (0.887 − 0.461i)37-s + (0.664 + 0.747i)41-s + (0.631 − 0.775i)43-s + (0.918 − 0.396i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $-0.988 + 0.153i$
Analytic conductor: \(24.0743\)
Root analytic conductor: \(24.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5184,\ (0:\ ),\ -0.988 + 0.153i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01991973821 - 0.2583070898i\)
\(L(\frac12)\) \(\approx\) \(0.01991973821 - 0.2583070898i\)
\(L(1)\) \(\approx\) \(0.8528576717 - 0.1387714651i\)
\(L(1)\) \(\approx\) \(0.8528576717 - 0.1387714651i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.585 - 0.810i)T \)
7 \( 1 + (-0.989 + 0.144i)T \)
11 \( 1 + (0.355 + 0.934i)T \)
13 \( 1 + (-0.756 - 0.653i)T \)
17 \( 1 + (-0.342 - 0.939i)T \)
19 \( 1 + (0.0436 + 0.999i)T \)
23 \( 1 + (-0.989 - 0.144i)T \)
29 \( 1 + (-0.561 + 0.827i)T \)
31 \( 1 + (0.396 - 0.918i)T \)
37 \( 1 + (0.887 - 0.461i)T \)
41 \( 1 + (0.664 + 0.747i)T \)
43 \( 1 + (0.631 - 0.775i)T \)
47 \( 1 + (0.918 - 0.396i)T \)
53 \( 1 + (-0.793 + 0.608i)T \)
59 \( 1 + (-0.409 - 0.912i)T \)
61 \( 1 + (-0.962 + 0.272i)T \)
67 \( 1 + (0.982 + 0.187i)T \)
71 \( 1 + (0.819 - 0.573i)T \)
73 \( 1 + (0.819 + 0.573i)T \)
79 \( 1 + (-0.998 + 0.0581i)T \)
83 \( 1 + (0.900 - 0.435i)T \)
89 \( 1 + (-0.573 + 0.819i)T \)
97 \( 1 + (-0.973 + 0.230i)T \)
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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

−18.45731652821871377414719229172, −17.46485871640236350801199633691, −17.17851604120478672883597205399, −16.3553948360027892373409209170, −15.68378444395778200925212255145, −15.02883231289125445475441039090, −14.14787421601572390860907327954, −13.827789217333284573719256170454, −13.09558651708690350064449431121, −12.39457383519045774534664299398, −11.53646245700401145148801451239, −10.89776268052312480361912864278, −10.29136641847560064220103585378, −9.42390629895936918205668425279, −9.218278572856444182177404965537, −8.09434686634706256517453800504, −7.31327996121672663241572339532, −6.437283298298674049325389589475, −6.28805855467464362185649182113, −5.43645773816976118673905074897, −4.28643916762114540091418825538, −3.69524374866869677159357296962, −2.77837610011037103697264220823, −2.32292296926879056310566891024, −1.21773151343981753297001271511, 0.069407557859665934012341446051, 1.12140074138361789828702943163, 2.17358051644608094932044386676, 2.64499088857249867579730887637, 3.80067629784991792713711466596, 4.43326670248114609820750510046, 5.291708534453060126049124675754, 5.884034489792839750805385294, 6.58521224293421194933467659404, 7.46033343830667354687659460989, 8.045446966813526969872878762077, 9.11893684923728992703015444375, 9.5929359993680085697808896619, 9.91041715982303485963279546273, 10.79976278370010259593086919846, 11.894232406662004917729643199615, 12.56363897764645035941787962104, 12.65358960730378977621692009155, 13.65811903091834449138861726284, 14.20442599888066253776914465293, 15.05511220554386421293399234855, 15.73245564421735992564627948232, 16.40381362107690802580255091724, 16.91257152412389332287068320451, 17.58253426504537609017536894664

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