Properties

Label 2-48-16.3-c8-0-22
Degree $2$
Conductor $48$
Sign $0.472 + 0.881i$
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  + (−15.6 + 3.51i)2-s + (−33.0 + 33.0i)3-s + (231. − 109. i)4-s + (801. − 801. i)5-s + (400. − 632. i)6-s + 3.98e3·7-s + (−3.22e3 + 2.52e3i)8-s − 2.18e3i·9-s + (−9.69e3 + 1.53e4i)10-s + (−1.18e4 − 1.18e4i)11-s + (−4.02e3 + 1.12e4i)12-s + (−9.80e3 − 9.80e3i)13-s + (−6.21e4 + 1.39e4i)14-s + 5.29e4i·15-s + (4.15e4 − 5.07e4i)16-s + 6.54e4·17-s + ⋯
L(s)  = 1  + (−0.975 + 0.219i)2-s + (−0.408 + 0.408i)3-s + (0.903 − 0.428i)4-s + (1.28 − 1.28i)5-s + (0.308 − 0.487i)6-s + 1.65·7-s + (−0.787 + 0.615i)8-s − 0.333i·9-s + (−0.969 + 1.53i)10-s + (−0.806 − 0.806i)11-s + (−0.194 + 0.543i)12-s + (−0.343 − 0.343i)13-s + (−1.61 + 0.363i)14-s + 1.04i·15-s + (0.633 − 0.773i)16-s + 0.783·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.472 + 0.881i$
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: \((2,\ 48,\ (\ :4),\ 0.472 + 0.881i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.22117 - 0.730992i\)
\(L(\frac12)\) \(\approx\) \(1.22117 - 0.730992i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (15.6 - 3.51i)T \)
3 \( 1 + (33.0 - 33.0i)T \)
good5 \( 1 + (-801. + 801. i)T - 3.90e5iT^{2} \)
7 \( 1 - 3.98e3T + 5.76e6T^{2} \)
11 \( 1 + (1.18e4 + 1.18e4i)T + 2.14e8iT^{2} \)
13 \( 1 + (9.80e3 + 9.80e3i)T + 8.15e8iT^{2} \)
17 \( 1 - 6.54e4T + 6.97e9T^{2} \)
19 \( 1 + (6.69e4 - 6.69e4i)T - 1.69e10iT^{2} \)
23 \( 1 - 1.71e5T + 7.83e10T^{2} \)
29 \( 1 + (8.41e5 + 8.41e5i)T + 5.00e11iT^{2} \)
31 \( 1 - 7.29e5iT - 8.52e11T^{2} \)
37 \( 1 + (7.22e5 - 7.22e5i)T - 3.51e12iT^{2} \)
41 \( 1 + 3.36e6iT - 7.98e12T^{2} \)
43 \( 1 + (-2.22e6 - 2.22e6i)T + 1.16e13iT^{2} \)
47 \( 1 + 6.58e6iT - 2.38e13T^{2} \)
53 \( 1 + (4.27e6 - 4.27e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (4.44e5 + 4.44e5i)T + 1.46e14iT^{2} \)
61 \( 1 + (-5.73e6 - 5.73e6i)T + 1.91e14iT^{2} \)
67 \( 1 + (-9.61e6 + 9.61e6i)T - 4.06e14iT^{2} \)
71 \( 1 - 3.11e7T + 6.45e14T^{2} \)
73 \( 1 + 8.30e6iT - 8.06e14T^{2} \)
79 \( 1 + 1.13e7iT - 1.51e15T^{2} \)
83 \( 1 + (-4.19e7 + 4.19e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 1.99e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.50e8T + 7.83e15T^{2} \)
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

−13.88034530150911854092935674604, −12.34136110077770039730213575719, −11.01792778367095374653023818512, −10.05127771594326709945939372732, −8.821845281103325594244255864528, −7.939850583063735837213791717435, −5.72059680960413601306922518525, −5.11429797098662339855062410349, −1.95282251488743155817264669822, −0.76857044440174660745769233835, 1.59422849408321662024753000317, 2.45181262102983506813323241002, 5.38117328218825964098624481836, 6.89728836605831688019704274510, 7.78677888255136731677510330125, 9.525227550470031493148939250040, 10.70012494636949951564879084685, 11.24655019008251617193740884266, 12.78902534975145170104035770249, 14.33190187220107906063198233642

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