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Properties

Label 1-6048-6048.1885-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.811 - 0.583i$
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  + (0.906 − 0.422i)5-s + (−0.422 + 0.906i)11-s + (0.573 − 0.819i)13-s − 17-s + (−0.707 − 0.707i)19-s + (−0.984 + 0.173i)23-s + (0.642 − 0.766i)25-s + (0.819 − 0.573i)29-s + (0.766 − 0.642i)31-s + (−0.258 − 0.965i)37-s + (−0.984 + 0.173i)41-s + (0.996 − 0.0871i)43-s + (−0.766 − 0.642i)47-s + (0.258 + 0.965i)53-s + i·55-s + ⋯
L(s)  = 1  + (0.906 − 0.422i)5-s + (−0.422 + 0.906i)11-s + (0.573 − 0.819i)13-s − 17-s + (−0.707 − 0.707i)19-s + (−0.984 + 0.173i)23-s + (0.642 − 0.766i)25-s + (0.819 − 0.573i)29-s + (0.766 − 0.642i)31-s + (−0.258 − 0.965i)37-s + (−0.984 + 0.173i)41-s + (0.996 − 0.0871i)43-s + (−0.766 − 0.642i)47-s + (0.258 + 0.965i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.811 - 0.583i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (1885, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.811 - 0.583i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3070833527 - 0.9531803829i\)
\(L(\frac12)\) \(\approx\) \(0.3070833527 - 0.9531803829i\)
\(L(1)\) \(\approx\) \(1.022304414 - 0.2053912952i\)
\(L(1)\) \(\approx\) \(1.022304414 - 0.2053912952i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.906 - 0.422i)T \)
11 \( 1 + (-0.422 + 0.906i)T \)
13 \( 1 + (0.573 - 0.819i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (-0.984 + 0.173i)T \)
29 \( 1 + (0.819 - 0.573i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (-0.984 + 0.173i)T \)
43 \( 1 + (0.996 - 0.0871i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (-0.819 - 0.573i)T \)
61 \( 1 + (0.0871 + 0.996i)T \)
67 \( 1 + (-0.422 - 0.906i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.819 + 0.573i)T \)
89 \( 1 - iT \)
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

−18.02526115908640047432832271974, −17.36641978633900857392121011981, −16.64421235075641675998073589711, −16.04557784990073432864970045504, −15.43496890548871101815164803790, −14.48281308647598379982625257012, −13.97504440061189119061194847731, −13.54677517523456333881702921484, −12.84607345992407480855744347444, −12.002687475110342977977175448487, −11.25838173699229776298222198000, −10.60317111813315587905877663948, −10.16217388872186724911572508952, −9.30966347855146108928897130884, −8.54613667474157769132906199953, −8.19201405138372054390075031253, −6.96038400848221101773547091534, −6.39335825995354930324828733581, −6.019701927740517982450378423688, −5.07959100838076555553268395501, −4.33610609536155672993421905088, −3.444470275423940747602809307760, −2.71415765301111889157122955127, −1.93000759962516206227907899590, −1.25322651654775949899884655262, 0.23016328539337607070911198181, 1.31210813479252553147449079862, 2.25657335912115361963584048096, 2.5637023615245133806110313449, 3.81595779654386544781714273908, 4.59443175574652458610061258806, 5.10419533751636556047149361401, 6.08943035533275756755928081770, 6.39349509373370703386366018088, 7.398760353815320030357974287414, 8.14464678637016579752933429323, 8.81798866843610339772552464444, 9.436353898187518419004857922141, 10.29262927696297725214791848290, 10.53195235476981762452685805426, 11.52870852718269808067116135532, 12.33517024666532470832599949686, 12.92491685651079548220344638099, 13.48776288339916088881106038931, 13.9441023644062740632335637189, 14.98334699259552218779199924127, 15.48712009341996513608780834202, 16.01881966203826779705555492971, 16.97391039569393463558177828301, 17.5868425928688116050877242791

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