Properties

Label 1-3024-3024.2147-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.318 + 0.947i$
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  + (−0.642 − 0.766i)5-s + (0.642 − 0.766i)11-s + (−0.342 + 0.939i)13-s + 17-s i·19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.342 + 0.939i)29-s + (0.173 + 0.984i)31-s + (0.866 + 0.5i)37-s + (0.939 + 0.342i)41-s + (0.984 + 0.173i)43-s + (−0.173 + 0.984i)47-s + (−0.866 − 0.5i)53-s − 55-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)5-s + (0.642 − 0.766i)11-s + (−0.342 + 0.939i)13-s + 17-s i·19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.342 + 0.939i)29-s + (0.173 + 0.984i)31-s + (0.866 + 0.5i)37-s + (0.939 + 0.342i)41-s + (0.984 + 0.173i)43-s + (−0.173 + 0.984i)47-s + (−0.866 − 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.318 + 0.947i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.318 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7983102236 + 1.110374579i\)
\(L(\frac12)\) \(\approx\) \(0.7983102236 + 1.110374579i\)
\(L(1)\) \(\approx\) \(0.9874222667 + 0.04170219036i\)
\(L(1)\) \(\approx\) \(0.9874222667 + 0.04170219036i\)

Euler product

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

−18.75384283651026249810955794389, −17.92431977610350784595876272185, −17.36244197868951857849037550297, −16.640302712151866081612167812561, −15.658492988733349747332034518050, −15.102825613087592472002290827292, −14.67720808284370441680183788990, −13.86387321212544792478236934564, −12.90390799086691990017665275977, −12.29503025538969895896367498024, −11.55571854715886102941362482558, −10.91431579239031193369364497875, −10.12676266536289751078980142289, −9.50065184488699099866481182893, −8.57976527555846648360509337451, −7.51899321899808537291474833478, −7.39796145195592784344238354901, −6.37194559359841238933097086457, −5.61385623341790089433307995283, −4.54250094501690294317544287653, −3.988949090103789384102865052145, −2.897917227527706043519909394613, −2.49230879403589316701692735703, −1.07438729425931220430911941393, −0.25512628893855260664743856542, 1.08995755131830857364043445526, 1.41630935904613582598353118446, 2.92246060862014456833249930158, 3.604070982428464312522380849812, 4.41458842358853175917746059533, 5.12403516962343539877543903198, 5.99162132042733569090751943156, 6.806189170127625347008513981405, 7.72057667405931876625434575760, 8.2659631104283027784115202108, 9.18622098063658470393630032267, 9.52873484155107056947472829606, 10.74293609373467998049824313638, 11.35322120548578013613843824067, 12.16725425225019482531894993963, 12.50144750779625204699631740309, 13.46597255904291135313101683243, 14.40783609378013021194611149848, 14.61483492067524762318912857383, 15.88458836943280493567583660254, 16.312319588015226931539074165764, 16.83359287421484373158719172476, 17.520583124347928339864033274429, 18.64053058239629702848540024018, 19.16751117679815307885572360282

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