↑

Properties

Label 2-48-16.11-c8-0-28
Degree $2$
Conductor $48$
Sign $-0.592 - 0.805i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−11.9 − 10.6i)2-s + (33.0 + 33.0i)3-s + (30.8 + 254. i)4-s + (−825. − 825. i)5-s + (−45.1 − 746. i)6-s + 1.80e3·7-s + (2.32e3 − 3.37e3i)8-s + 2.18e3i·9-s + (1.12e3 + 1.86e4i)10-s + (1.16e4 − 1.16e4i)11-s + (−7.38e3 + 9.42e3i)12-s + (−1.35e4 + 1.35e4i)13-s + (−2.15e4 − 1.91e4i)14-s − 5.45e4i·15-s + (−6.36e4 + 1.56e4i)16-s − 3.29e3·17-s + ⋯
L(s)  = 1  + (−0.748 − 0.663i)2-s + (0.408 + 0.408i)3-s + (0.120 + 0.992i)4-s + (−1.32 − 1.32i)5-s + (−0.0348 − 0.576i)6-s + 0.751·7-s + (0.568 − 0.822i)8-s + 0.333i·9-s + (0.112 + 1.86i)10-s + (0.794 − 0.794i)11-s + (−0.356 + 0.454i)12-s + (−0.474 + 0.474i)13-s + (−0.562 − 0.498i)14-s − 1.07i·15-s + (−0.970 + 0.239i)16-s − 0.0393·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.592 - 0.805i$
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: \((2,\ 48,\ (\ :4),\ -0.592 - 0.805i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.00256093 + 0.00506303i\)
\(L(\frac12)\) \(\approx\) \(0.00256093 + 0.00506303i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (11.9 + 10.6i)T \)
3 \( 1 + (-33.0 - 33.0i)T \)
good5 \( 1 + (825. + 825. i)T + 3.90e5iT^{2} \)
7 \( 1 - 1.80e3T + 5.76e6T^{2} \)
11 \( 1 + (-1.16e4 + 1.16e4i)T - 2.14e8iT^{2} \)
13 \( 1 + (1.35e4 - 1.35e4i)T - 8.15e8iT^{2} \)
17 \( 1 + 3.29e3T + 6.97e9T^{2} \)
19 \( 1 + (7.96e4 + 7.96e4i)T + 1.69e10iT^{2} \)
23 \( 1 + 3.29e5T + 7.83e10T^{2} \)
29 \( 1 + (8.99e5 - 8.99e5i)T - 5.00e11iT^{2} \)
31 \( 1 - 9.52e5iT - 8.52e11T^{2} \)
37 \( 1 + (7.93e5 + 7.93e5i)T + 3.51e12iT^{2} \)
41 \( 1 + 1.67e5iT - 7.98e12T^{2} \)
43 \( 1 + (-2.53e6 + 2.53e6i)T - 1.16e13iT^{2} \)
47 \( 1 - 7.95e6iT - 2.38e13T^{2} \)
53 \( 1 + (7.64e6 + 7.64e6i)T + 6.22e13iT^{2} \)
59 \( 1 + (-3.94e6 + 3.94e6i)T - 1.46e14iT^{2} \)
61 \( 1 + (1.74e7 - 1.74e7i)T - 1.91e14iT^{2} \)
67 \( 1 + (-8.13e6 - 8.13e6i)T + 4.06e14iT^{2} \)
71 \( 1 + 1.48e6T + 6.45e14T^{2} \)
73 \( 1 - 1.20e7iT - 8.06e14T^{2} \)
79 \( 1 + 4.86e7iT - 1.51e15T^{2} \)
83 \( 1 + (-2.80e6 - 2.80e6i)T + 2.25e15iT^{2} \)
89 \( 1 + 1.06e8iT - 3.93e15T^{2} \)
97 \( 1 + 3.18e7T + 7.83e15T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.76256740541152471158617218208, −11.79738237795474460983089226640, −10.97077994902902116052470574291, −9.135230245411321001081824813661, −8.580822986627245827029427202551, −7.51644674511606871826609209146, −4.65680732357470740286091867897, −3.66772704925475066071011282281, −1.50127110543632188488979142018, −0.00256787196017994902227775905, 2.07962296851316899374443863228, 4.10906992614680589136250517782, 6.37391755416831596973217896596, 7.57165951799446393477576229972, 8.043894459665314437250062250979, 9.805984885544515212691611911071, 11.07397994530759911589146193173, 12.03630592224459536487526229585, 14.14164140348116442618120692844, 14.92816730693462065023962261219

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