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Properties

Label 1-5328-5328.3947-r0-0-0
Degree $1$
Conductor $5328$
Sign $-0.944 - 0.329i$
Analytic cond. $24.7431$
Root an. cond. $24.7431$
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)5-s + (0.173 + 0.984i)7-s + (0.866 − 0.5i)11-s + (−0.984 + 0.173i)13-s + (−0.939 + 0.342i)17-s + (0.984 − 0.173i)19-s + (−0.5 + 0.866i)23-s + (−0.766 − 0.642i)25-s + (−0.866 + 0.5i)29-s + (−0.5 − 0.866i)31-s + (0.984 + 0.173i)35-s + (0.173 + 0.984i)41-s + (−0.866 − 0.5i)43-s + 47-s + (−0.939 + 0.342i)49-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)5-s + (0.173 + 0.984i)7-s + (0.866 − 0.5i)11-s + (−0.984 + 0.173i)13-s + (−0.939 + 0.342i)17-s + (0.984 − 0.173i)19-s + (−0.5 + 0.866i)23-s + (−0.766 − 0.642i)25-s + (−0.866 + 0.5i)29-s + (−0.5 − 0.866i)31-s + (0.984 + 0.173i)35-s + (0.173 + 0.984i)41-s + (−0.866 − 0.5i)43-s + 47-s + (−0.939 + 0.342i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5328\)    =    \(2^{4} \cdot 3^{2} \cdot 37\)
Sign: $-0.944 - 0.329i$
Analytic conductor: \(24.7431\)
Root analytic conductor: \(24.7431\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5328} (3947, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5328,\ (0:\ ),\ -0.944 - 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07332653111 - 0.4326837496i\)
\(L(\frac12)\) \(\approx\) \(0.07332653111 - 0.4326837496i\)
\(L(1)\) \(\approx\) \(0.9346746151 - 0.1004169546i\)
\(L(1)\) \(\approx\) \(0.9346746151 - 0.1004169546i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( 1 \)
good5 \( 1 + (0.342 - 0.939i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (0.984 - 0.173i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + T \)
53 \( 1 + (0.642 - 0.766i)T \)
59 \( 1 + (-0.984 - 0.173i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 - T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.984 - 0.173i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 - T \)
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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

−18.11818681356029459812376033722, −17.526853333987675326004932357025, −17.09813110717128513350822850110, −16.330781381270118243295839160967, −15.48507849338773449441802294970, −14.755970754324348651382196164910, −14.24803780673137367352819936105, −13.80697189609904590842607886778, −13.00163885694345395993390435376, −12.13847367047046268695236269989, −11.51111732238009520686923860616, −10.80190733954259784137099601988, −10.17232450508351986626722682516, −9.64524988527367652203036404463, −8.9296644136659144099162252545, −7.83842181129794226761015751878, −7.08140575350399384344424956124, −6.97408316600544622245637941833, −5.99674095134172109653286496105, −5.13477651823912105520804704571, −4.27746827603064204915136751273, −3.72605640726866356035341336112, −2.76252506808406451835821303017, −2.07642061875071447625710930035, −1.19688105866777757459652503205, 0.10659617794121353211861119552, 1.42318593677227130503069491912, 1.93549884055461069915617825266, 2.82231266060299162539564523056, 3.824452628645881123402168679758, 4.54815534965093276397762870159, 5.40550744913835789577436458040, 5.735396506434752430926379468635, 6.636611236846252467536780695374, 7.50663864197195063997996633516, 8.28987714538104750860675761677, 9.026637557507895725175857668890, 9.36208136139217735125522906023, 9.994940830280347404944761833116, 11.250886825564794909627785963777, 11.648941145482267248669700899623, 12.27135943041805897377197443542, 12.95841768539661264883936798968, 13.6260888335911645638097629411, 14.295083455222066560247652419274, 15.10232647740080497943979334145, 15.58227801793902133281548130778, 16.53034842179916390105407888066, 16.83344564250019606148812212627, 17.67756803357672116346078460647

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