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Properties

Label 2-48-12.11-c21-0-11
Degree $2$
Conductor $48$
Sign $0.796 - 0.604i$
Analytic cond. $134.149$
Root an. cond. $11.5822$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.14e4 − 6.18e4i)3-s − 1.06e7i·5-s + 3.89e8i·7-s + (2.80e9 − 1.00e10i)9-s − 1.23e11·11-s − 7.26e11·13-s + (−6.55e11 − 8.63e11i)15-s − 1.04e13i·17-s + 4.84e13i·19-s + (2.40e13 + 3.17e13i)21-s − 2.08e13·23-s + 3.64e14·25-s + (−3.95e14 − 9.94e14i)27-s − 1.94e15i·29-s + 4.62e15i·31-s + ⋯
L(s)  = 1  + (0.796 − 0.604i)3-s − 0.485i·5-s + 0.521i·7-s + (0.267 − 0.963i)9-s − 1.43·11-s − 1.46·13-s + (−0.293 − 0.386i)15-s − 1.25i·17-s + 1.81i·19-s + (0.315 + 0.414i)21-s − 0.104·23-s + 0.764·25-s + (−0.369 − 0.929i)27-s − 0.857i·29-s + 1.01i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.796 - 0.604i$
Analytic conductor: \(134.149\)
Root analytic conductor: \(11.5822\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :21/2),\ 0.796 - 0.604i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.684120899\)
\(L(\frac12)\) \(\approx\) \(1.684120899\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-8.14e4 + 6.18e4i)T \)
good5 \( 1 + 1.06e7iT - 4.76e14T^{2} \)
7 \( 1 - 3.89e8iT - 5.58e17T^{2} \)
11 \( 1 + 1.23e11T + 7.40e21T^{2} \)
13 \( 1 + 7.26e11T + 2.47e23T^{2} \)
17 \( 1 + 1.04e13iT - 6.90e25T^{2} \)
19 \( 1 - 4.84e13iT - 7.14e26T^{2} \)
23 \( 1 + 2.08e13T + 3.94e28T^{2} \)
29 \( 1 + 1.94e15iT - 5.13e30T^{2} \)
31 \( 1 - 4.62e15iT - 2.08e31T^{2} \)
37 \( 1 - 3.28e16T + 8.55e32T^{2} \)
41 \( 1 - 1.13e17iT - 7.38e33T^{2} \)
43 \( 1 + 7.69e16iT - 2.00e34T^{2} \)
47 \( 1 - 5.40e17T + 1.30e35T^{2} \)
53 \( 1 - 1.83e18iT - 1.62e36T^{2} \)
59 \( 1 + 5.25e17T + 1.54e37T^{2} \)
61 \( 1 + 4.40e18T + 3.10e37T^{2} \)
67 \( 1 + 2.76e17iT - 2.22e38T^{2} \)
71 \( 1 + 2.91e19T + 7.52e38T^{2} \)
73 \( 1 - 3.60e19T + 1.34e39T^{2} \)
79 \( 1 - 9.16e19iT - 7.08e39T^{2} \)
83 \( 1 - 9.06e19T + 1.99e40T^{2} \)
89 \( 1 + 5.08e20iT - 8.65e40T^{2} \)
97 \( 1 + 2.06e20T + 5.27e41T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.05722301391977281031287619235, −10.20297180726184199571883974715, −9.228551391399705960407781748543, −8.041915324142820640351402052790, −7.35143596618230184085339307937, −5.74899642823441566942726566190, −4.61955586387068845091929096091, −2.92273649876244727901334854263, −2.26726253695461325655052001654, −0.882321715478251033736630984069, 0.34073183570965852059840964404, 2.25656035022269657199847551577, 2.89991976325119559878078537077, 4.27812169174098412224061270029, 5.24492622417524673956021960633, 7.07551994516542381510245122059, 7.87964320123458679241315517121, 9.171954761912841329750381527316, 10.32673273866581943555614513148, 10.89240222740835137108945027562

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