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Properties

Label 1-3024-3024.2077-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.206 + 0.978i$
Analytic cond. $324.973$
Root an. cond. $324.973$
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 + (−0.342 + 0.939i)11-s + (0.342 + 0.939i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.173 + 0.984i)23-s + (−0.766 + 0.642i)25-s + (0.342 − 0.939i)29-s + (0.939 − 0.342i)31-s − i·37-s + (−0.939 + 0.342i)41-s + (0.984 − 0.173i)43-s + (0.939 + 0.342i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)5-s + (−0.342 + 0.939i)11-s + (0.342 + 0.939i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.173 + 0.984i)23-s + (−0.766 + 0.642i)25-s + (0.342 − 0.939i)29-s + (0.939 − 0.342i)31-s − i·37-s + (−0.939 + 0.342i)41-s + (0.984 − 0.173i)43-s + (0.939 + 0.342i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.206 + 0.978i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.206 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.075612873 + 1.326007915i\)
\(L(\frac12)\) \(\approx\) \(1.075612873 + 1.326007915i\)
\(L(1)\) \(\approx\) \(1.024381616 + 0.1066811863i\)
\(L(1)\) \(\approx\) \(1.024381616 + 0.1066811863i\)

Euler product

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

−18.54628413418719543179481467988, −18.241467014308154226259070984430, −17.38742607281960237253442831102, −16.45996354433020762850932533245, −15.72358966489515478629907745270, −15.40364946988425557043488036702, −14.25995995239172850210766824612, −13.97606246074348249611398485153, −13.15009949735930131688011619978, −12.17514265017749693519118163703, −11.58884968374136493937235238025, −10.743508300875568734440000379987, −10.376715969453532060324471198614, −9.46912283581240765338635589474, −8.42698117530429440517178795135, −7.95808095628665321457575441395, −7.0601017908485633150393522047, −6.44705579410466171591437085394, −5.53061655302581269861192968553, −4.87195682410772990517048799095, −3.65317793512637583704011361538, −3.04660952774506268716455909559, −2.54267081410559247273572713160, −1.03175658577691542796508151858, −0.33141363068777479297327771431, 0.98255938848188210385712381837, 1.63579174336808579361393385458, 2.60276554884268604968680257989, 3.982696631832929458495712146162, 4.12893422893946739619299784113, 5.26901697536166268279242294276, 5.80815514836130336457406858973, 6.861578811466677151098666071539, 7.74342925093352692932535794606, 8.15463242077555395019090826869, 9.20814383577058375911336018001, 9.6419837340147936017032588943, 10.48680347789028430818871932685, 11.4766363764560039348273964805, 12.14622694806250436305722132023, 12.516872505357211721250936946194, 13.580693889911706414798699968497, 13.931217608686627458868737750630, 15.136805113546344113532988651629, 15.52775643003169491493098035844, 16.32536443465971673914605287597, 16.95717221224234033835073571418, 17.567227985834645034264350614381, 18.387707053546125055765535859296, 19.19988864927711159672829870777

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