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Properties

Label 1-72e2-5184.709-r0-0-0
Degree $1$
Conductor $5184$
Sign $-0.556 - 0.830i$
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  + (0.244 + 0.969i)5-s + (0.979 + 0.202i)7-s + (0.487 − 0.873i)11-s + (−0.631 − 0.775i)13-s + (−0.984 − 0.173i)17-s + (−0.537 − 0.843i)19-s + (−0.979 + 0.202i)23-s + (−0.880 + 0.474i)25-s + (−0.409 + 0.912i)29-s + (0.0581 − 0.998i)31-s + (0.0436 + 0.999i)35-s + (0.999 + 0.0436i)37-s + (0.851 − 0.524i)41-s + (0.999 − 0.0145i)43-s + (−0.998 + 0.0581i)47-s + ⋯
L(s)  = 1  + (0.244 + 0.969i)5-s + (0.979 + 0.202i)7-s + (0.487 − 0.873i)11-s + (−0.631 − 0.775i)13-s + (−0.984 − 0.173i)17-s + (−0.537 − 0.843i)19-s + (−0.979 + 0.202i)23-s + (−0.880 + 0.474i)25-s + (−0.409 + 0.912i)29-s + (0.0581 − 0.998i)31-s + (0.0436 + 0.999i)35-s + (0.999 + 0.0436i)37-s + (0.851 − 0.524i)41-s + (0.999 − 0.0145i)43-s + (−0.998 + 0.0581i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $-0.556 - 0.830i$
Analytic conductor: \(24.0743\)
Root analytic conductor: \(24.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5184,\ (0:\ ),\ -0.556 - 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3391049060 - 0.6353465025i\)
\(L(\frac12)\) \(\approx\) \(0.3391049060 - 0.6353465025i\)
\(L(1)\) \(\approx\) \(0.9923051317 + 0.001179081561i\)
\(L(1)\) \(\approx\) \(0.9923051317 + 0.001179081561i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.244 + 0.969i)T \)
7 \( 1 + (0.979 + 0.202i)T \)
11 \( 1 + (0.487 - 0.873i)T \)
13 \( 1 + (-0.631 - 0.775i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
19 \( 1 + (-0.537 - 0.843i)T \)
23 \( 1 + (-0.979 + 0.202i)T \)
29 \( 1 + (-0.409 + 0.912i)T \)
31 \( 1 + (0.0581 - 0.998i)T \)
37 \( 1 + (0.999 + 0.0436i)T \)
41 \( 1 + (0.851 - 0.524i)T \)
43 \( 1 + (0.999 - 0.0145i)T \)
47 \( 1 + (-0.998 + 0.0581i)T \)
53 \( 1 + (-0.608 - 0.793i)T \)
59 \( 1 + (-0.962 - 0.272i)T \)
61 \( 1 + (-0.0726 + 0.997i)T \)
67 \( 1 + (-0.934 - 0.355i)T \)
71 \( 1 + (-0.0871 - 0.996i)T \)
73 \( 1 + (-0.0871 + 0.996i)T \)
79 \( 1 + (0.230 + 0.973i)T \)
83 \( 1 + (-0.810 + 0.585i)T \)
89 \( 1 + (0.996 + 0.0871i)T \)
97 \( 1 + (-0.597 + 0.802i)T \)
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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

−17.87962077733645854974535533167, −17.59735784691301780686984408667, −16.94425415944854065152435558853, −16.36773707774372781563752888114, −15.59740289499475352312041123150, −14.7122907434729840077106860934, −14.34492761880903792837343845628, −13.59091851993852135800632520369, −12.78646020511950215527180402264, −12.18877205355608001406467737589, −11.67918410201853116531639526782, −10.86714142770318055854487121910, −10.03170139198407645990261026677, −9.37347784434778640733323191709, −8.8152301130574562078027651497, −7.95976252262308848798201206277, −7.529460722799220963945647276538, −6.451083743539784726535366394903, −5.90791719681235495424677178700, −4.787491538015351192482963781038, −4.45729438436932875872862015395, −3.97633061657527641689398996789, −2.41382923084762808293415640687, −1.84495238672593961435039278842, −1.27193676397366085217296638555, 0.170906197501636873797481296546, 1.43490693642384267018193640098, 2.36994605077011683578443151382, 2.787692956612206935041144741226, 3.84915803732023906091408808520, 4.52515388756521536177005717464, 5.46673645714190005663956625795, 6.05720396724224801916590006287, 6.78135820198538071597026450073, 7.60134906701626349163439876930, 8.120640096437535394190031840680, 9.02117625571572650567381335169, 9.60091057962404141552442901818, 10.58339636864829891542246067184, 11.11318406339178627513769398989, 11.434317486602051625111945120384, 12.38159252128371669550165655489, 13.23134060495602423646665048094, 13.87134519349517948232772131613, 14.504001725654602094627226600523, 15.01316501663109523742670356022, 15.580045022623255414423210679405, 16.480445507859196414408061770753, 17.31260082774930069002783303275, 17.85622765813528152087543057423

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