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Properties

Label 1-3024-3024.1579-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.604 + 0.796i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.604 + 0.796i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.604 + 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6565110876 + 1.322596817i\)
\(L(\frac12)\) \(\approx\) \(0.6565110876 + 1.322596817i\)
\(L(1)\) \(\approx\) \(1.080411499 + 0.1093283332i\)
\(L(1)\) \(\approx\) \(1.080411499 + 0.1093283332i\)

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

−18.66335118276223429155253125760, −17.92273299249383606613195839067, −17.4076061079013456649947311467, −16.50943672468529374185951827563, −15.80045073740753584438908246296, −15.109182355693209797217483924849, −14.32263320194979495601461644230, −13.81873626417352093076568142906, −13.18368085069952410659320066345, −12.13124330147074192201981659162, −11.52766070912384947579951189168, −10.749179538458902631499439139293, −10.17826200077458423225804104734, −9.42662034540003089243423744212, −8.598360337369681729542763719132, −7.696103197660325821918106344640, −7.12239358000849217309408630466, −6.12054290175127782637362233567, −5.76422264903132567265106580325, −4.69415108411653670710573891690, −3.73647907462654971416313762842, −2.833856317358159502142730776499, −2.465054490953913341971618338427, −1.072455765998380231858474248, −0.242814662622753457397279581310, 1.16368977696622962355690662919, 1.66355565006949023320827996533, 2.63255047901248464573338471336, 3.7871354110193800003059913725, 4.60820957509968096337128312193, 5.05237043748212959575336319271, 6.05939487477533673448477386894, 6.85295481111305496440248772492, 7.54193005589630495252865401996, 8.57396680165665573963178027916, 9.10300349208610596468459213450, 9.77092361365939730984412250698, 10.43268881439879567352086620168, 11.67677932319690664806178528257, 11.93370263798916578076486063753, 12.84378907267901478377832304243, 13.4474175329484102722081268792, 14.123766723278349827561198690032, 14.99150522752884785142632961412, 15.65512836569910131150272750112, 16.40975235451514209186138701296, 17.115058281184390039713936797406, 17.5939222863194186442373732828, 18.2562418719040636081711769008, 19.39998116003001741484618685771

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