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Properties

Label 1-3024-3024.1075-r1-0-0
Degree $1$
Conductor $3024$
Sign $0.377 - 0.925i$
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  + (0.984 − 0.173i)5-s + (0.984 + 0.173i)11-s + (−0.984 + 0.173i)13-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.766 − 0.642i)23-s + (0.939 − 0.342i)25-s + (0.984 + 0.173i)29-s + (−0.173 − 0.984i)31-s − i·37-s + (−0.173 − 0.984i)41-s + (−0.642 + 0.766i)43-s + (−0.173 + 0.984i)47-s + (−0.866 + 0.5i)53-s + 55-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)5-s + (0.984 + 0.173i)11-s + (−0.984 + 0.173i)13-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.766 − 0.642i)23-s + (0.939 − 0.342i)25-s + (0.984 + 0.173i)29-s + (−0.173 − 0.984i)31-s − i·37-s + (−0.173 − 0.984i)41-s + (−0.642 + 0.766i)43-s + (−0.173 + 0.984i)47-s + (−0.866 + 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.377 - 0.925i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.377 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.927527376 - 1.295190885i\)
\(L(\frac12)\) \(\approx\) \(1.927527376 - 1.295190885i\)
\(L(1)\) \(\approx\) \(1.232749288 - 0.09253630403i\)
\(L(1)\) \(\approx\) \(1.232749288 - 0.09253630403i\)

Euler product

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

−19.059978041928401713317454101863, −18.2010900079797558428316438128, −17.43590219732561609716933941404, −17.165564617855774949772954722131, −16.35594620488002291094497547212, −15.404061128111735844811962534533, −14.69369069038231162234127128174, −14.15490627344388489126585072687, −13.37251919214864524854263324270, −12.8444907021406845512959606289, −11.802026053408391880449432658235, −11.350013594869318578120432532352, −10.232543634152214424840747975337, −9.880078426514769858936504247, −8.943223963285821838100841917, −8.53724198908204183994671226323, −7.07729903760770776743654163029, −6.86930924914197608224808938508, −5.979685448978560722043551967821, −5.0348705909121201261450436635, −4.55759065729358409735326471998, −3.27000357891880072368021290776, −2.626082242400492545243759962565, −1.7358812525989188636372923861, −0.855384831392715509183413036977, 0.38792029179551711212165216944, 1.56550876534339257278708918661, 2.10779873070340508499177783012, 3.035756953254973921142967530972, 4.22827017680710847935134487248, 4.696606063087057278179654227429, 5.76207068137627704736932280247, 6.39984118182024822927164896913, 6.97266017378303068720809053092, 8.04952700067642544839938204819, 8.9009579862088301196209513679, 9.3787632350812073939973405341, 10.21325764232295456677250320199, 10.77829807787614970566713830993, 11.740923418215473569366897435, 12.62872484441793682345867815265, 12.90555428033035206893785956326, 13.9617885292750969955169500321, 14.58635808309834030543588829080, 14.95106986373383148525323340797, 16.11158686282280433957888372263, 16.887284043225575282662252951194, 17.30934272738327449465890408246, 17.78923402824789194465504623866, 18.90271787808065638260656235986

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