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Properties

Label 1-6048-6048.461-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.0943 - 0.995i$
Analytic cond. $28.0867$
Root an. cond. $28.0867$
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  + (−0.573 − 0.819i)5-s + (0.819 + 0.573i)11-s + (0.906 + 0.422i)13-s + (0.5 − 0.866i)17-s + (−0.258 + 0.965i)19-s + (0.642 − 0.766i)23-s + (−0.342 + 0.939i)25-s + (−0.422 − 0.906i)29-s + (−0.766 − 0.642i)31-s + (0.258 + 0.965i)37-s + (−0.342 − 0.939i)41-s + (−0.819 − 0.573i)43-s + (−0.766 + 0.642i)47-s + (0.707 − 0.707i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.573 − 0.819i)5-s + (0.819 + 0.573i)11-s + (0.906 + 0.422i)13-s + (0.5 − 0.866i)17-s + (−0.258 + 0.965i)19-s + (0.642 − 0.766i)23-s + (−0.342 + 0.939i)25-s + (−0.422 − 0.906i)29-s + (−0.766 − 0.642i)31-s + (0.258 + 0.965i)37-s + (−0.342 − 0.939i)41-s + (−0.819 − 0.573i)43-s + (−0.766 + 0.642i)47-s + (0.707 − 0.707i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0943 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0943 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.0943 - 0.995i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.0943 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.123355251 - 1.021876006i\)
\(L(\frac12)\) \(\approx\) \(1.123355251 - 1.021876006i\)
\(L(1)\) \(\approx\) \(1.011742751 - 0.2074357942i\)
\(L(1)\) \(\approx\) \(1.011742751 - 0.2074357942i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.573 - 0.819i)T \)
11 \( 1 + (0.819 + 0.573i)T \)
13 \( 1 + (0.906 + 0.422i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.258 + 0.965i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (-0.422 - 0.906i)T \)
31 \( 1 + (-0.766 - 0.642i)T \)
37 \( 1 + (0.258 + 0.965i)T \)
41 \( 1 + (-0.342 - 0.939i)T \)
43 \( 1 + (-0.819 - 0.573i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (-0.573 - 0.819i)T \)
61 \( 1 + (0.996 + 0.0871i)T \)
67 \( 1 + (0.906 + 0.422i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (-0.422 - 0.906i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
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

−18.08035201652763997158046113177, −17.171098479247391854494415949589, −16.49839579330359344232419892741, −15.90276960138457993702881167685, −15.046868946566630363468029273353, −14.78396398145758905116027971097, −14.004500306478978778726471799158, −13.237399059857223049177780424007, −12.68222324809429981534577172133, −11.70884439562485198574947840695, −11.215356089370524949257858093935, −10.761229610221291260583791615215, −10.02496695592412742515242500836, −9.01644503374635187714434083880, −8.59414746047414486846171967881, −7.76006610545998778542411216154, −7.065056495206984897402472167387, −6.441681497717780013694281679667, −5.79139688601764824103581121823, −4.94522820013791664525191434309, −3.859788947036555684780237111224, −3.50476505255087799517189736604, −2.85772929505161938315996478179, −1.69574207600034599668912053677, −0.937050135213605253267754544867, 0.4566370938166065766983282523, 1.39126890590569597380413437696, 2.01674838916006596569926870932, 3.26467676467376554794482918253, 3.909598279770167838913252180376, 4.4732016153175634709932030480, 5.25522385519849215436117320836, 6.02688482250653930932534023619, 6.8334091611945275988004634352, 7.4839450064295719473994432519, 8.33408634905421817998133434180, 8.76220978537254136882888116612, 9.573410137874938586570280298378, 10.07253327220507332195311072318, 11.23808590263122320426936435508, 11.58371409978276518372745458526, 12.268499811852826277163095431986, 12.8851427666764272994442439174, 13.55233106123203233330555583308, 14.343000572440577642840103469, 14.95286809288984331685039465544, 15.63133765258453406517064898434, 16.40643724165044855167462438853, 16.75865627545343207010754031939, 17.34396433716199033537798946433

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