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Properties

Label 1-6048-6048.475-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.531 + 0.847i$
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  + (−0.0871 − 0.996i)5-s + (0.996 + 0.0871i)11-s + (0.573 + 0.819i)13-s + (−0.5 + 0.866i)17-s + (−0.965 − 0.258i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (−0.819 − 0.573i)29-s + (−0.939 + 0.342i)31-s + (−0.965 + 0.258i)37-s + (0.984 + 0.173i)41-s + (0.996 + 0.0871i)43-s + (0.939 + 0.342i)47-s + (−0.707 − 0.707i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.0871 − 0.996i)5-s + (0.996 + 0.0871i)11-s + (0.573 + 0.819i)13-s + (−0.5 + 0.866i)17-s + (−0.965 − 0.258i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (−0.819 − 0.573i)29-s + (−0.939 + 0.342i)31-s + (−0.965 + 0.258i)37-s + (0.984 + 0.173i)41-s + (0.996 + 0.0871i)43-s + (0.939 + 0.342i)47-s + (−0.707 − 0.707i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.531 + 0.847i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.531 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.088418016 + 0.6022856539i\)
\(L(\frac12)\) \(\approx\) \(1.088418016 + 0.6022856539i\)
\(L(1)\) \(\approx\) \(0.9916998340 - 0.04135583257i\)
\(L(1)\) \(\approx\) \(0.9916998340 - 0.04135583257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.0871 - 0.996i)T \)
11 \( 1 + (0.996 + 0.0871i)T \)
13 \( 1 + (0.573 + 0.819i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.965 - 0.258i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (-0.819 - 0.573i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (0.984 + 0.173i)T \)
43 \( 1 + (0.996 + 0.0871i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.0871 + 0.996i)T \)
61 \( 1 + (-0.906 + 0.422i)T \)
67 \( 1 + (-0.573 - 0.819i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.819 + 0.573i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
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

−17.63995250474502877622297174807, −17.08156500157697019217023460525, −16.20444609690168552325068674092, −15.5036373372239827513049132484, −15.073509917732771841452605600859, −14.17524576745507039245045223607, −13.9461125994519489693601786674, −12.97850738691607568433679347541, −12.34423310747792927864073508241, −11.495525544082141191277355525937, −10.94023855437139029271085617213, −10.54977544230752169101587600347, −9.48622832813099386174280446720, −9.10265627686085053404848198006, −8.16485938185089186650861264845, −7.37143590772490542349692847308, −6.94337085660186548515705261810, −5.996917413913550876084906234507, −5.67606958491576485154672331890, −4.4739409207356668570241968450, −3.71278017608289767760554092738, −3.26766343388041481930375478415, −2.272247735629585232652436946340, −1.59558059803519352506868164184, −0.34780226607550870295340687091, 0.90437925693360377435573716562, 1.7367249157701202947724953426, 2.28338286243119744165532268221, 3.72227190598314187646141264410, 4.114747074072518729913618609968, 4.66310970687933067996193319276, 5.72205750082163413895092104970, 6.29013730381532071306080799177, 6.93591269663204226419423101563, 7.92097788870239169598521590327, 8.605518287860477240645285347310, 9.09059940323123513048339021953, 9.57180019091327210894194779052, 10.790286407134919948369592542311, 11.05318313377165295322916890940, 12.14233089461747293687244446959, 12.40804913767524770689043073210, 13.182786390339480006901333786491, 13.83574586889544593373245495204, 14.543460994896558915793145418604, 15.21064984682323995385370051383, 15.94377655831822324828392884627, 16.63282703301591258361114000018, 17.019712156775126922839888599531, 17.63488533093627913705225802419

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