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Properties

Label 1-3024-3024.2981-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  + (−0.642 + 0.766i)5-s + (−0.642 − 0.766i)11-s + (0.984 + 0.173i)13-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (−0.984 + 0.173i)29-s + (0.939 − 0.342i)31-s + (−0.866 − 0.5i)37-s + (−0.173 + 0.984i)41-s + (0.642 + 0.766i)43-s + (−0.939 − 0.342i)47-s − i·53-s + 55-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)5-s + (−0.642 − 0.766i)11-s + (0.984 + 0.173i)13-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (−0.984 + 0.173i)29-s + (0.939 − 0.342i)31-s + (−0.866 − 0.5i)37-s + (−0.173 + 0.984i)41-s + (0.642 + 0.766i)43-s + (−0.939 − 0.342i)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: $\chi_{3024} (2981, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ -0.827 - 0.561i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03172185713 - 0.1032203353i\)
\(L(\frac12)\) \(\approx\) \(0.03172185713 - 0.1032203353i\)
\(L(1)\) \(\approx\) \(0.7745186598 + 0.07842802893i\)
\(L(1)\) \(\approx\) \(0.7745186598 + 0.07842802893i\)

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 \( 1 + (0.984 + 0.173i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.984 + 0.173i)T \)
31 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.642 + 0.766i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 - iT \)
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 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.984 - 0.173i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−19.339723149203225796970135341077, −18.578567111727221892647589834477, −17.99841463555833780861476137337, −17.265064931195318764721460328919, −16.37326117989711571919217037785, −15.689573034321847091322664310150, −15.560903310351701294614703902638, −14.38890354194904104591010276858, −13.57506375066014012030201114455, −13.06898491799969242516960742909, −12.07638458273833474069167285215, −11.8440179912235410280630784294, −10.8042773902502208198792212239, −10.11774857663267070040176352757, −9.22571291045759077575445562278, −8.611589531598716801267557912203, −7.76667487025430546601538325104, −7.31987354645800685573700805043, −6.223332104844194605642697185764, −5.37147784301945839604816624558, −4.70591590266689926617104291913, −3.93727070262099709530097490602, −3.13631856817144536152791621825, −2.04053847026281045846644683283, −1.14084966688219752049560785027, 0.03515408484091886027072419996, 1.362657176488075080932865794386, 2.42570026276105877186721744403, 3.33478132470433803656808824952, 3.80417567604588475280743334047, 4.78750925177633125715748752783, 5.862782307285605142613664428283, 6.34704131377198802448463503782, 7.27256288973863992205372624278, 8.04116193087898583567235933144, 8.51353097484181782169681415115, 9.542838774663052146628721657585, 10.38094092238609738614647065388, 11.12178175036809397620466074659, 11.423568595973306482752562976826, 12.35475648574703658461301668824, 13.380085544109101782774306975361, 13.697669555207326533299902557955, 14.66867929207047788865263591056, 15.31216876794389932370394905746, 16.020113282266339496574897529799, 16.37080226637533160647975632753, 17.645532131337546027747553439210, 18.0706492500656419695950840167, 18.806314008404159074347748914138

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