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Properties

Label 1-6048-6048.187-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.986 - 0.162i$
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 + (0.422 − 0.906i)13-s + (−0.5 − 0.866i)17-s + (0.258 + 0.965i)19-s + (0.984 + 0.173i)23-s + (0.642 + 0.766i)25-s + (0.906 − 0.422i)29-s + (−0.939 + 0.342i)31-s + (0.707 + 0.707i)37-s + (−0.342 − 0.939i)41-s + (−0.573 + 0.819i)43-s + (0.939 + 0.342i)47-s + (0.965 − 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.906 + 0.422i)5-s + (−0.422 − 0.906i)11-s + (0.422 − 0.906i)13-s + (−0.5 − 0.866i)17-s + (0.258 + 0.965i)19-s + (0.984 + 0.173i)23-s + (0.642 + 0.766i)25-s + (0.906 − 0.422i)29-s + (−0.939 + 0.342i)31-s + (0.707 + 0.707i)37-s + (−0.342 − 0.939i)41-s + (−0.573 + 0.819i)43-s + (0.939 + 0.342i)47-s + (0.965 − 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.986 - 0.162i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.986 - 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.294363576 - 0.1875344292i\)
\(L(\frac12)\) \(\approx\) \(2.294363576 - 0.1875344292i\)
\(L(1)\) \(\approx\) \(1.304464145 + 0.02496750570i\)
\(L(1)\) \(\approx\) \(1.304464145 + 0.02496750570i\)

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 \( 1 + (0.422 - 0.906i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (0.906 - 0.422i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.342 - 0.939i)T \)
43 \( 1 + (-0.573 + 0.819i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (0.965 - 0.258i)T \)
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 \( 1 - iT \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.906 + 0.422i)T \)
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

−17.64429311466836121415344844679, −17.14419672707359933821960854339, −16.49269983440337527956462591337, −15.81018293221665088161174097229, −15.01703049015823955443946183879, −14.52915636952069110426081617724, −13.53378616645364648415277578620, −13.271199886164628096079958180490, −12.56919300907334937867382177016, −11.8854803607240171446944302312, −10.938021162807359788915352851250, −10.50267754827454089493451734182, −9.638475996623367373858502667183, −9.04560810385523181778206571850, −8.63553959136233204447961007236, −7.58183246327609413264020534414, −6.831236804560954358597551935448, −6.32628109360376701493224423090, −5.433468789076637270441415235477, −4.77552798481571634971867806558, −4.23613732987501254798990958823, −3.1596106139500412430235390947, −2.2211423547405615999087387087, −1.78702420647523662649658489033, −0.79859868415112894262356756474, 0.74872635783929559280698758003, 1.51181572698607694381506934784, 2.66624524883819011757500764523, 2.96388649520962590512117054432, 3.84512924861345310277736603785, 4.96711748516150197247868569003, 5.56008717190413378968897990417, 6.06850125045458420786394732967, 6.88059813899384177425741238102, 7.58155014115325042863933625679, 8.42600722916813543978933124091, 8.99598037574640928234788524935, 9.82477291445523230168335108317, 10.41291537594030542788523868857, 10.97881247284103240791928998314, 11.61385906993582648300987029760, 12.59415803812313411101337699682, 13.20527306508863803901822150984, 13.7872913978766731706182722160, 14.21439450787345355656083013299, 15.1748613073626835611963120402, 15.609373311148232274760197229657, 16.56138697722459427113926575862, 16.903469198801071206504837469860, 17.949740966549309643075999562949

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