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Properties

Label 1-3024-3024.2459-r0-0-0
Degree $1$
Conductor $3024$
Sign $-0.135 + 0.990i$
Analytic cond. $14.0433$
Root an. cond. $14.0433$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.135 + 0.990i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ -0.135 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.078019290 + 1.235708783i\)
\(L(\frac12)\) \(\approx\) \(1.078019290 + 1.235708783i\)
\(L(1)\) \(\approx\) \(1.090411822 + 0.3216523665i\)
\(L(1)\) \(\approx\) \(1.090411822 + 0.3216523665i\)

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

−18.78094717123677794906491420890, −18.05934119873735048765933690907, −17.43386236698473691721673011950, −16.84244832245139718128213971578, −16.055031357972491909417067691793, −15.530521429209012147530475121457, −14.50122801505273674399772231599, −14.07746741273141962297652427936, −12.998179410376528879836196118256, −12.56040604537497547425420319772, −12.06895775472955880796552111300, −11.0080302113006242954435376768, −10.14168835616879226779660979212, −9.70404736116240062813230781731, −8.70810204275509292890906537089, −8.27977286633804082010701483893, −7.33692041102741936069582925333, −6.4912106527262949851867503636, −5.68001547185930835946660752105, −4.90914070781036305570727348176, −4.33286999765275386505643417144, −3.31465317764637669068630893906, −2.30837220596190150388208300592, −1.47634311043074935283978590378, −0.52382497622523056574751965002, 1.15924367835307475308013710196, 2.04813928165334734025300459600, 2.9329054178362744485109868746, 3.71287129516920358359049828006, 4.4214343342473015203671581967, 5.658317291837802835396682601628, 6.27105868861243757039471619221, 6.731747888287328964734100194751, 7.765341087072580741315219369841, 8.4847323198064121761121083944, 9.27344245117212162624694354387, 10.09858442439077288593938318316, 10.688652039221028238074798461386, 11.54233440802533468132223869097, 11.87378946841716344019197040946, 13.19786527184849366019927949147, 13.6572069092795474327468632861, 14.24841415901302119103053458582, 15.10533461023613811619214708136, 15.544950805536503209915282071148, 16.67525202950441899432184931116, 17.06895937874888351745369962727, 17.86088650971069474531934862349, 18.68056277676472598283879107474, 19.244149635627242023244337529250

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