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Properties

Label 1-3024-3024.2491-r0-0-0
Degree $1$
Conductor $3024$
Sign $-0.827 - 0.561i$
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 + (−0.642 + 0.766i)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 + (−0.642 − 0.766i)29-s + (0.173 − 0.984i)31-s + (−0.866 − 0.5i)37-s + (0.766 + 0.642i)41-s + (−0.342 + 0.939i)43-s + (0.173 + 0.984i)47-s + i·53-s − 55-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)5-s + (−0.342 + 0.939i)11-s + (−0.642 + 0.766i)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 + (−0.642 − 0.766i)29-s + (0.173 − 0.984i)31-s + (−0.866 − 0.5i)37-s + (0.766 + 0.642i)41-s + (−0.342 + 0.939i)43-s + (0.173 + 0.984i)47-s + i·53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.827 - 0.561i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ -0.827 - 0.561i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06212369090 + 0.2021454222i\)
\(L(\frac12)\) \(\approx\) \(-0.06212369090 + 0.2021454222i\)
\(L(1)\) \(\approx\) \(0.8324600192 + 0.2258709540i\)
\(L(1)\) \(\approx\) \(0.8324600192 + 0.2258709540i\)

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 \( 1 + (-0.642 + 0.766i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.642 - 0.766i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + iT \)
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 \( 1 + (-0.642 - 0.766i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−18.66304122293893238673901480814, −17.79209725415393610151973380705, −17.11423562209015535800145145128, −16.749870995013591820951966800555, −15.81617100045369924562241195141, −15.2628543283759172425922044204, −14.3901984767525584329812612517, −13.56542262985148608115801954179, −13.01318987092455153969898452467, −12.39980550071954205068421922732, −11.68317023880996797029721187528, −10.56856631866572663036014309149, −10.294610728121807004159226960456, −9.136825319234528630216947054050, −8.67953205847823696254906027527, −7.96528822868417589887757572666, −7.14078666137280633567701194013, −6.08061351287450939824804096353, −5.42234082991223279593599050598, −4.931369849893265279683685908565, −3.81255729049254896256734217087, −3.097469204812505209502150298247, −2.01354527828493747989863799483, −1.19053993766068927009152995684, −0.06119778879684357445486355066, 1.61198050474868653299395193678, 2.38730118027339309949875848493, 2.94625399475814080091926953247, 4.18772213980809952685693553501, 4.68550036268564145019034032536, 5.839566864007739938699398701369, 6.40541892575150527380021222837, 7.37519072771189911944356001001, 7.61825164342222028579891469224, 8.8662297515307536231805441932, 9.716966125613344453620342159236, 10.075678620670719371515150133395, 10.960458497821648351802792792219, 11.63754681900520168037243948099, 12.444079436518034344015639495048, 13.104930354956200515553097577873, 14.07753016302634064660514967483, 14.548279446886424002624880121072, 15.09621975515907961316717817636, 15.9336559518539425820144727584, 16.83390811161636536630185408610, 17.34512145093546920410500264731, 18.17601620960232024941723696753, 18.78155052249232334101723324148, 19.21141998562712815369032746635

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