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Properties

Label 1-3024-3024.2363-r0-0-0
Degree $1$
Conductor $3024$
Sign $-0.796 - 0.604i$
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

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

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.796 - 0.604i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ -0.796 - 0.604i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08314338714 - 0.2470398738i\)
\(L(\frac12)\) \(\approx\) \(0.08314338714 - 0.2470398738i\)
\(L(1)\) \(\approx\) \(0.8158465543 + 0.04351029475i\)
\(L(1)\) \(\approx\) \(0.8158465543 + 0.04351029475i\)

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

−19.476782618351518974835941516171, −18.65409643570838573024022892016, −17.666383678092021309795954524734, −17.427869607212189317426389684368, −16.428658646481791557588014218087, −15.92879112313432179611378342231, −15.09905695496303487061598805768, −14.65346398061685970681874705336, −13.47332577493257704299174874137, −12.9695568992548262719120588082, −12.26295027491376733961270141574, −11.804585820234429592099650148450, −10.76759737379846581995822418729, −9.88726562123813673655047132301, −9.541180305819059242839577998823, −8.38811946856940337950543471379, −7.89634391411415547591153017972, −7.3005171155717751065536785063, −6.17255991950269027817659056428, −5.205779936640038177583105419072, −4.96726633645431361196250006942, −3.78564570761987524105914337386, −3.21632911537677886856137106117, −1.90175275178049681831812344457, −1.2595042801208783641519795650, 0.08083771920407574763285109486, 1.34577235807751342787190128261, 2.688157152971630806483634091708, 2.94303708773540704150507415628, 3.996263232822819270488178100734, 4.82082429837179837850100639431, 5.71370250167683243047672916279, 6.62417977057567501706365250467, 7.08187077448830928488063500452, 7.993352013570789177510148248095, 8.61278472999907094248771685657, 9.70548605976899961363610396336, 10.14437689084747825259072971180, 11.13401821958982485255379328656, 11.64716066588279196456693534979, 12.13826181716728450355625215631, 13.49985936786597706593777839849, 13.78581654398368044976189191830, 14.544993443291841125113364802956, 15.257699903925482159445578110081, 16.04978528406885941774354561420, 16.52138908837527479214086871501, 17.429446406988559216003440772757, 18.30684749532791432280451194258, 18.77996507107304369413027342834

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