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Properties

Label 2-48-16.3-c8-0-14
Degree $2$
Conductor $48$
Sign $0.740 - 0.671i$
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  + (−1.76 + 15.9i)2-s + (33.0 − 33.0i)3-s + (−249. − 56.0i)4-s + (47.4 − 47.4i)5-s + (467. + 584. i)6-s − 2.24e3·7-s + (1.33e3 − 3.87e3i)8-s − 2.18e3i·9-s + (670. + 838. i)10-s + (1.78e4 + 1.78e4i)11-s + (−1.01e4 + 6.40e3i)12-s + (−1.69e4 − 1.69e4i)13-s + (3.95e3 − 3.57e4i)14-s − 3.13e3i·15-s + (5.92e4 + 2.79e4i)16-s + 1.41e5·17-s + ⋯
L(s)  = 1  + (−0.110 + 0.993i)2-s + (0.408 − 0.408i)3-s + (−0.975 − 0.218i)4-s + (0.0759 − 0.0759i)5-s + (0.360 + 0.450i)6-s − 0.935·7-s + (0.324 − 0.945i)8-s − 0.333i·9-s + (0.0670 + 0.0838i)10-s + (1.22 + 1.22i)11-s + (−0.487 + 0.309i)12-s + (−0.593 − 0.593i)13-s + (0.103 − 0.930i)14-s − 0.0619i·15-s + (0.904 + 0.427i)16-s + 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.740 - 0.671i$
Analytic conductor: \(19.5541\)
Root analytic conductor: \(4.42201\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :4),\ 0.740 - 0.671i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.70355 + 0.657659i\)
\(L(\frac12)\) \(\approx\) \(1.70355 + 0.657659i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.76 - 15.9i)T \)
3 \( 1 + (-33.0 + 33.0i)T \)
good5 \( 1 + (-47.4 + 47.4i)T - 3.90e5iT^{2} \)
7 \( 1 + 2.24e3T + 5.76e6T^{2} \)
11 \( 1 + (-1.78e4 - 1.78e4i)T + 2.14e8iT^{2} \)
13 \( 1 + (1.69e4 + 1.69e4i)T + 8.15e8iT^{2} \)
17 \( 1 - 1.41e5T + 6.97e9T^{2} \)
19 \( 1 + (-1.65e5 + 1.65e5i)T - 1.69e10iT^{2} \)
23 \( 1 - 6.49e4T + 7.83e10T^{2} \)
29 \( 1 + (-6.71e4 - 6.71e4i)T + 5.00e11iT^{2} \)
31 \( 1 - 1.30e6iT - 8.52e11T^{2} \)
37 \( 1 + (-2.03e6 + 2.03e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 1.65e6iT - 7.98e12T^{2} \)
43 \( 1 + (-1.66e6 - 1.66e6i)T + 1.16e13iT^{2} \)
47 \( 1 + 9.60e6iT - 2.38e13T^{2} \)
53 \( 1 + (-1.35e6 + 1.35e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (4.17e6 + 4.17e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (-5.53e6 - 5.53e6i)T + 1.91e14iT^{2} \)
67 \( 1 + (-1.60e7 + 1.60e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 1.51e7T + 6.45e14T^{2} \)
73 \( 1 - 9.31e6iT - 8.06e14T^{2} \)
79 \( 1 + 4.17e7iT - 1.51e15T^{2} \)
83 \( 1 + (-1.25e7 + 1.25e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 4.99e7iT - 3.93e15T^{2} \)
97 \( 1 + 4.98e7T + 7.83e15T^{2} \)
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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

−14.30035016286256841485670939913, −13.06430811168766004969104358485, −12.18573151847001583877545705986, −9.820126334096645110636921060057, −9.252489660409983574779324350829, −7.54998376370476150847808074836, −6.78872460484128646940329721050, −5.21454183170506524207283277949, −3.38639509463013399921609759523, −0.975862995031568941706440110489, 1.00707479077752409887639489281, 2.97739367167228887360383888039, 3.93428574313731585790196374577, 5.87738319789288394420143484880, 7.975902175504558468925238601044, 9.427384160164590697056519033828, 9.937642780802407496704058097315, 11.50294225741151687392212789939, 12.41043302843130414214552138991, 13.91484947565885362174214249354

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