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Properties

Label 2-48-16.11-c8-0-24
Degree $2$
Conductor $48$
Sign $0.951 + 0.308i$
Analytic cond. $19.5541$
Root an. cond. $4.42201$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.42 + 15.6i)2-s + (33.0 + 33.0i)3-s + (−232. + 107. i)4-s + (−488. − 488. i)5-s + (−403. + 630. i)6-s − 90.0·7-s + (−2.47e3 − 3.26e3i)8-s + 2.18e3i·9-s + (5.95e3 − 9.31e3i)10-s + (5.12e3 − 5.12e3i)11-s + (−1.12e4 − 4.14e3i)12-s + (2.34e4 − 2.34e4i)13-s + (−308. − 1.40e3i)14-s − 3.23e4i·15-s + (4.25e4 − 4.98e4i)16-s + 2.93e4·17-s + ⋯
L(s)  = 1  + (0.214 + 0.976i)2-s + (0.408 + 0.408i)3-s + (−0.908 + 0.418i)4-s + (−0.781 − 0.781i)5-s + (−0.311 + 0.486i)6-s − 0.0375·7-s + (−0.603 − 0.797i)8-s + 0.333i·9-s + (0.595 − 0.931i)10-s + (0.349 − 0.349i)11-s + (−0.541 − 0.199i)12-s + (0.821 − 0.821i)13-s + (−0.00804 − 0.0366i)14-s − 0.638i·15-s + (0.649 − 0.760i)16-s + 0.351·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.951 + 0.308i$
Analytic conductor: \(19.5541\)
Root analytic conductor: \(4.42201\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :4),\ 0.951 + 0.308i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.38680 - 0.219470i\)
\(L(\frac12)\) \(\approx\) \(1.38680 - 0.219470i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.42 - 15.6i)T \)
3 \( 1 + (-33.0 - 33.0i)T \)
good5 \( 1 + (488. + 488. i)T + 3.90e5iT^{2} \)
7 \( 1 + 90.0T + 5.76e6T^{2} \)
11 \( 1 + (-5.12e3 + 5.12e3i)T - 2.14e8iT^{2} \)
13 \( 1 + (-2.34e4 + 2.34e4i)T - 8.15e8iT^{2} \)
17 \( 1 - 2.93e4T + 6.97e9T^{2} \)
19 \( 1 + (3.09e4 + 3.09e4i)T + 1.69e10iT^{2} \)
23 \( 1 + 3.12e5T + 7.83e10T^{2} \)
29 \( 1 + (-8.20e5 + 8.20e5i)T - 5.00e11iT^{2} \)
31 \( 1 + 3.53e5iT - 8.52e11T^{2} \)
37 \( 1 + (-1.50e6 - 1.50e6i)T + 3.51e12iT^{2} \)
41 \( 1 + 1.63e6iT - 7.98e12T^{2} \)
43 \( 1 + (8.04e5 - 8.04e5i)T - 1.16e13iT^{2} \)
47 \( 1 + 8.02e6iT - 2.38e13T^{2} \)
53 \( 1 + (9.84e6 + 9.84e6i)T + 6.22e13iT^{2} \)
59 \( 1 + (6.21e6 - 6.21e6i)T - 1.46e14iT^{2} \)
61 \( 1 + (-5.08e6 + 5.08e6i)T - 1.91e14iT^{2} \)
67 \( 1 + (1.21e7 + 1.21e7i)T + 4.06e14iT^{2} \)
71 \( 1 - 1.42e6T + 6.45e14T^{2} \)
73 \( 1 - 5.27e6iT - 8.06e14T^{2} \)
79 \( 1 - 2.02e6iT - 1.51e15T^{2} \)
83 \( 1 + (-1.30e7 - 1.30e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 6.83e7iT - 3.93e15T^{2} \)
97 \( 1 - 4.03e7T + 7.83e15T^{2} \)
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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

−14.01427527029558029444590782597, −12.99829878105024176884797091703, −11.77127352708348107236504358822, −9.932181156726993109922311469423, −8.514058450426054227941977173239, −7.984711374709145885909165162015, −6.14254167679824837712351777159, −4.64548266373581315217062372234, −3.53440850619183093079931024669, −0.49461728123552683636138681146, 1.46936580407843246747747533448, 3.07862668748881951627803278498, 4.22033854996828019178767494397, 6.39614485066265393937670095907, 7.956703281227796523370055620306, 9.274014195243573737492120320864, 10.67933416645399638890667942857, 11.68729265460036065402276530679, 12.63073531872902324312257085865, 14.02892503510458243117724612296

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