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Properties

Label 2-48-16.3-c8-0-16
Degree $2$
Conductor $48$
Sign $0.999 - 0.0135i$
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  + (14.0 − 7.59i)2-s + (−33.0 + 33.0i)3-s + (140. − 213. i)4-s + (−341. + 341. i)5-s + (−214. + 716. i)6-s + 1.35e3·7-s + (360. − 4.08e3i)8-s − 2.18e3i·9-s + (−2.21e3 + 7.40e3i)10-s + (7.12e3 + 7.12e3i)11-s + (2.41e3 + 1.17e4i)12-s + (2.39e4 + 2.39e4i)13-s + (1.90e4 − 1.02e4i)14-s − 2.25e4i·15-s + (−2.58e4 − 6.02e4i)16-s + 1.42e5·17-s + ⋯
L(s)  = 1  + (0.880 − 0.474i)2-s + (−0.408 + 0.408i)3-s + (0.549 − 0.835i)4-s + (−0.546 + 0.546i)5-s + (−0.165 + 0.553i)6-s + 0.562·7-s + (0.0879 − 0.996i)8-s − 0.333i·9-s + (−0.221 + 0.740i)10-s + (0.486 + 0.486i)11-s + (0.116 + 0.565i)12-s + (0.837 + 0.837i)13-s + (0.495 − 0.266i)14-s − 0.446i·15-s + (−0.395 − 0.918i)16-s + 1.70·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0135i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.999 - 0.0135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.999 - 0.0135i$
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: \((2,\ 48,\ (\ :4),\ 0.999 - 0.0135i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.94623 + 0.0199366i\)
\(L(\frac12)\) \(\approx\) \(2.94623 + 0.0199366i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-14.0 + 7.59i)T \)
3 \( 1 + (33.0 - 33.0i)T \)
good5 \( 1 + (341. - 341. i)T - 3.90e5iT^{2} \)
7 \( 1 - 1.35e3T + 5.76e6T^{2} \)
11 \( 1 + (-7.12e3 - 7.12e3i)T + 2.14e8iT^{2} \)
13 \( 1 + (-2.39e4 - 2.39e4i)T + 8.15e8iT^{2} \)
17 \( 1 - 1.42e5T + 6.97e9T^{2} \)
19 \( 1 + (1.68e4 - 1.68e4i)T - 1.69e10iT^{2} \)
23 \( 1 - 2.84e5T + 7.83e10T^{2} \)
29 \( 1 + (4.12e4 + 4.12e4i)T + 5.00e11iT^{2} \)
31 \( 1 + 4.92e5iT - 8.52e11T^{2} \)
37 \( 1 + (-1.44e5 + 1.44e5i)T - 3.51e12iT^{2} \)
41 \( 1 + 3.50e6iT - 7.98e12T^{2} \)
43 \( 1 + (-4.26e6 - 4.26e6i)T + 1.16e13iT^{2} \)
47 \( 1 - 2.41e6iT - 2.38e13T^{2} \)
53 \( 1 + (7.04e6 - 7.04e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (4.76e6 + 4.76e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (1.12e7 + 1.12e7i)T + 1.91e14iT^{2} \)
67 \( 1 + (1.80e7 - 1.80e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 5.35e6T + 6.45e14T^{2} \)
73 \( 1 - 5.24e6iT - 8.06e14T^{2} \)
79 \( 1 - 1.76e7iT - 1.51e15T^{2} \)
83 \( 1 + (-2.65e7 + 2.65e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 5.13e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.36e8T + 7.83e15T^{2} \)
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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

−14.14721402002613251849084707769, −12.54586750742095339869188002647, −11.51670774984153845102052188342, −10.84289339309235090100812504279, −9.459631941794940733306239965704, −7.38484913541895046106937612545, −5.98189735334924877803615078778, −4.52417360669563198193703149680, −3.36809161747499179208655156434, −1.33705245404056650083949766183, 1.07835465936570654988916244641, 3.37905395746100516989991759103, 4.92799805973548362975488757048, 6.06088000964717582236503046387, 7.60779083553652720778757786285, 8.510894979434081092705069534724, 10.86367857120546767961617256456, 11.90781600729996768127754143108, 12.73488260486749766507145018793, 13.89607820329158460707712458951

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