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Properties

Label 2-48-16.13-c5-0-18
Degree $2$
Conductor $48$
Sign $-0.844 - 0.535i$
Analytic cond. $7.69842$
Root an. cond. $2.77460$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.12 − 2.38i)2-s + (−6.36 − 6.36i)3-s + (20.5 + 24.4i)4-s + (12.6 − 12.6i)5-s + (17.4 + 47.8i)6-s − 213. i·7-s + (−47.1 − 174. i)8-s + 81i·9-s + (−94.9 + 34.6i)10-s + (−416. + 416. i)11-s + (24.7 − 286. i)12-s + (377. + 377. i)13-s + (−509. + 1.09e3i)14-s − 160.·15-s + (−175. + 1.00e3i)16-s − 2.25e3·17-s + ⋯
L(s)  = 1  + (−0.906 − 0.422i)2-s + (−0.408 − 0.408i)3-s + (0.643 + 0.765i)4-s + (0.225 − 0.225i)5-s + (0.197 + 0.542i)6-s − 1.64i·7-s + (−0.260 − 0.965i)8-s + 0.333i·9-s + (−0.300 + 0.109i)10-s + (−1.03 + 1.03i)11-s + (0.0496 − 0.575i)12-s + (0.619 + 0.619i)13-s + (−0.695 + 1.49i)14-s − 0.184·15-s + (−0.171 + 0.985i)16-s − 1.89·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.844 - 0.535i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ -0.844 - 0.535i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0609448 + 0.209988i\)
\(L(\frac12)\) \(\approx\) \(0.0609448 + 0.209988i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.12 + 2.38i)T \)
3 \( 1 + (6.36 + 6.36i)T \)
good5 \( 1 + (-12.6 + 12.6i)T - 3.12e3iT^{2} \)
7 \( 1 + 213. iT - 1.68e4T^{2} \)
11 \( 1 + (416. - 416. i)T - 1.61e5iT^{2} \)
13 \( 1 + (-377. - 377. i)T + 3.71e5iT^{2} \)
17 \( 1 + 2.25e3T + 1.41e6T^{2} \)
19 \( 1 + (758. + 758. i)T + 2.47e6iT^{2} \)
23 \( 1 + 218. iT - 6.43e6T^{2} \)
29 \( 1 + (2.26e3 + 2.26e3i)T + 2.05e7iT^{2} \)
31 \( 1 + 5.25e3T + 2.86e7T^{2} \)
37 \( 1 + (-722. + 722. i)T - 6.93e7iT^{2} \)
41 \( 1 - 7.89e3iT - 1.15e8T^{2} \)
43 \( 1 + (-1.63e4 + 1.63e4i)T - 1.47e8iT^{2} \)
47 \( 1 + 8.06e3T + 2.29e8T^{2} \)
53 \( 1 + (1.19e4 - 1.19e4i)T - 4.18e8iT^{2} \)
59 \( 1 + (-1.32e4 + 1.32e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (-1.53e4 - 1.53e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (3.94e4 + 3.94e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 3.84e4iT - 1.80e9T^{2} \)
73 \( 1 + 1.85e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.72e3T + 3.07e9T^{2} \)
83 \( 1 + (-3.31e4 - 3.31e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 7.18e4iT - 5.58e9T^{2} \)
97 \( 1 + 7.73e4T + 8.58e9T^{2} \)
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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

−13.44671948837501606161874208100, −12.89542993234414199714154667279, −11.18309817467152289353278319018, −10.57458897945682477415475508179, −9.193296985865236823515320079302, −7.59388177579536092275347031834, −6.74927549096044368573497327573, −4.30917318013702965851071332747, −1.88318385226991429694147844408, −0.14910935272292284527200282257, 2.47869485680236331582272841998, 5.46101480276614022985973986510, 6.26687939024800820108458111550, 8.299421795146868306706945511632, 9.098177635725136398104871533786, 10.60600512549350661488416995228, 11.34422673376728206840976689454, 12.94196583353939763958511133487, 14.68299192080127760603136646757, 15.72256895802786414792906518214

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