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Properties

Label 2-48-16.11-c4-0-0
Degree $2$
Conductor $48$
Sign $-0.712 - 0.701i$
Analytic cond. $4.96175$
Root an. cond. $2.22750$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.09 − 2.53i)2-s + (3.67 + 3.67i)3-s + (3.18 + 15.6i)4-s + (−7.37 − 7.37i)5-s + (−2.07 − 20.6i)6-s − 75.1·7-s + (29.8 − 56.6i)8-s + 27i·9-s + (4.16 + 41.5i)10-s + (−55.7 + 55.7i)11-s + (−45.9 + 69.3i)12-s + (−199. + 199. i)13-s + (232. + 190. i)14-s − 54.2i·15-s + (−235. + 99.7i)16-s + 153.·17-s + ⋯
L(s)  = 1  + (−0.774 − 0.632i)2-s + (0.408 + 0.408i)3-s + (0.198 + 0.980i)4-s + (−0.295 − 0.295i)5-s + (−0.0576 − 0.574i)6-s − 1.53·7-s + (0.466 − 0.884i)8-s + 0.333i·9-s + (0.0416 + 0.415i)10-s + (−0.461 + 0.461i)11-s + (−0.318 + 0.481i)12-s + (−1.18 + 1.18i)13-s + (1.18 + 0.971i)14-s − 0.240i·15-s + (−0.920 + 0.389i)16-s + 0.531·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.712 - 0.701i$
Analytic conductor: \(4.96175\)
Root analytic conductor: \(2.22750\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :2),\ -0.712 - 0.701i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0981241 + 0.239441i\)
\(L(\frac12)\) \(\approx\) \(0.0981241 + 0.239441i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.09 + 2.53i)T \)
3 \( 1 + (-3.67 - 3.67i)T \)
good5 \( 1 + (7.37 + 7.37i)T + 625iT^{2} \)
7 \( 1 + 75.1T + 2.40e3T^{2} \)
11 \( 1 + (55.7 - 55.7i)T - 1.46e4iT^{2} \)
13 \( 1 + (199. - 199. i)T - 2.85e4iT^{2} \)
17 \( 1 - 153.T + 8.35e4T^{2} \)
19 \( 1 + (-1.48 - 1.48i)T + 1.30e5iT^{2} \)
23 \( 1 + 693.T + 2.79e5T^{2} \)
29 \( 1 + (-1.01e3 + 1.01e3i)T - 7.07e5iT^{2} \)
31 \( 1 - 1.12e3iT - 9.23e5T^{2} \)
37 \( 1 + (-55.0 - 55.0i)T + 1.87e6iT^{2} \)
41 \( 1 - 16.6iT - 2.82e6T^{2} \)
43 \( 1 + (213. - 213. i)T - 3.41e6iT^{2} \)
47 \( 1 - 1.36e3iT - 4.87e6T^{2} \)
53 \( 1 + (1.48e3 + 1.48e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (-723. + 723. i)T - 1.21e7iT^{2} \)
61 \( 1 + (-2.59e3 + 2.59e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (-4.63e3 - 4.63e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 8.62e3T + 2.54e7T^{2} \)
73 \( 1 - 8.79e3iT - 2.83e7T^{2} \)
79 \( 1 - 5.38e3iT - 3.89e7T^{2} \)
83 \( 1 + (6.98e3 + 6.98e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 8.18e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.48e4T + 8.85e7T^{2} \)
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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

−15.87370158648869189297736281056, −14.07885564523557242045614296464, −12.67446758067315066970088512566, −11.92875389236706175135240246397, −10.04701743432162109791807889026, −9.684930157963555611088830043281, −8.249648932616654306527700889920, −6.83107662034156377360950629256, −4.21082278282798866375938749757, −2.61463817055329053506338587864, 0.18199308386345331974781389725, 2.95007276384222920229903773919, 5.75756153982101482133502350812, 7.07878432951449353989289695162, 8.088382620643846182888790452061, 9.580778396709954683248131726339, 10.42219076470829242705775515274, 12.25935682281417572888388147187, 13.45785499109789116381238352288, 14.75163067022662014196375676725

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