Properties

Label 1-3024-3024.1357-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.982 + 0.187i$
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 + (0.642 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.982 + 0.187i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.982 + 0.187i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01036959014 - 0.1093592602i\)
\(L(\frac12)\) \(\approx\) \(0.01036959014 - 0.1093592602i\)
\(L(1)\) \(\approx\) \(0.8941869445 - 0.1018327844i\)
\(L(1)\) \(\approx\) \(0.8941869445 - 0.1018327844i\)

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

−19.06510122717547308267969954156, −18.738829270028534954324713918205, −18.00157870151515043624241522615, −17.08660605389044389187599230323, −16.45960922655276578655068601443, −15.69001673027556512456211093693, −15.06703075903836192173444067409, −14.31088382834765851225215854477, −13.77469408952191187607089816438, −12.95144859428732275441595713442, −12.13018179415301756797594823416, −11.25049340350963715954004271371, −10.849596880752541955574784890906, −10.20285275712062448719154847695, −9.19239722855096292502218503520, −8.36853068209437284963169542758, −7.85562147629905294076666417199, −6.8481690510619441858209665629, −6.21088723541058151858036485029, −5.68058313600686103346446017626, −4.189046121852786454288496163196, −3.980538521446812180333397317088, −2.795570280056762395602055683278, −2.273696105192913210989087863360, −0.97554553942820398560482919424, 0.020190925834827238936046516865, 1.06392365928379880448038364727, 1.77679919617244722713149408772, 2.96717307568440895254235237597, 3.73383373672347719197935408976, 4.71614768391290670035376170513, 5.163819286567533266727555791011, 6.05328693942881666117717619702, 7.02088409450052520665536968062, 7.884134038354327143555807657969, 8.298016587447701838686707765695, 9.23878239851956329464885664713, 9.88964477556841775797481227051, 10.63530566758819140818625613027, 11.58103995155687230920438956484, 12.18214564326131476547595533651, 12.95061534737175441269948955716, 13.292064083368042194339778297394, 14.41616915106332109013599385792, 15.03417637144227063012450060365, 15.94467965714500648288080950179, 16.19585558789488299541425456796, 17.12505099128424603328542296712, 17.87479806539350479842354520541, 18.31774446901281778599860023978

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