↑

Properties

Label 2-48-16.3-c8-0-13
Degree $2$
Conductor $48$
Sign $0.0265 - 0.999i$
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  + (1.46 + 15.9i)2-s + (33.0 − 33.0i)3-s + (−251. + 46.6i)4-s + (−173. + 173. i)5-s + (575. + 478. i)6-s + 3.86e3·7-s + (−1.11e3 − 3.94e3i)8-s − 2.18e3i·9-s + (−3.01e3 − 2.50e3i)10-s + (−6.95e3 − 6.95e3i)11-s + (−6.78e3 + 9.86e3i)12-s + (2.91e4 + 2.91e4i)13-s + (5.65e3 + 6.16e4i)14-s + 1.14e4i·15-s + (6.11e4 − 2.34e4i)16-s + 7.58e4·17-s + ⋯
L(s)  = 1  + (0.0914 + 0.995i)2-s + (0.408 − 0.408i)3-s + (−0.983 + 0.182i)4-s + (−0.277 + 0.277i)5-s + (0.443 + 0.369i)6-s + 1.61·7-s + (−0.271 − 0.962i)8-s − 0.333i·9-s + (−0.301 − 0.250i)10-s + (−0.475 − 0.475i)11-s + (−0.327 + 0.475i)12-s + (1.01 + 1.01i)13-s + (0.147 + 1.60i)14-s + 0.226i·15-s + (0.933 − 0.357i)16-s + 0.908·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0265 - 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.0265 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.0265 - 0.999i$
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: \((2,\ 48,\ (\ :4),\ 0.0265 - 0.999i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.63770 + 1.59474i\)
\(L(\frac12)\) \(\approx\) \(1.63770 + 1.59474i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.46 - 15.9i)T \)
3 \( 1 + (-33.0 + 33.0i)T \)
good5 \( 1 + (173. - 173. i)T - 3.90e5iT^{2} \)
7 \( 1 - 3.86e3T + 5.76e6T^{2} \)
11 \( 1 + (6.95e3 + 6.95e3i)T + 2.14e8iT^{2} \)
13 \( 1 + (-2.91e4 - 2.91e4i)T + 8.15e8iT^{2} \)
17 \( 1 - 7.58e4T + 6.97e9T^{2} \)
19 \( 1 + (1.64e5 - 1.64e5i)T - 1.69e10iT^{2} \)
23 \( 1 - 2.49e5T + 7.83e10T^{2} \)
29 \( 1 + (-7.68e5 - 7.68e5i)T + 5.00e11iT^{2} \)
31 \( 1 + 3.58e5iT - 8.52e11T^{2} \)
37 \( 1 + (2.21e6 - 2.21e6i)T - 3.51e12iT^{2} \)
41 \( 1 + 2.80e6iT - 7.98e12T^{2} \)
43 \( 1 + (-6.55e5 - 6.55e5i)T + 1.16e13iT^{2} \)
47 \( 1 + 2.55e6iT - 2.38e13T^{2} \)
53 \( 1 + (-1.00e7 + 1.00e7i)T - 6.22e13iT^{2} \)
59 \( 1 + (-6.95e6 - 6.95e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (1.15e6 + 1.15e6i)T + 1.91e14iT^{2} \)
67 \( 1 + (-1.74e7 + 1.74e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 6.78e5T + 6.45e14T^{2} \)
73 \( 1 - 5.75e6iT - 8.06e14T^{2} \)
79 \( 1 - 2.53e6iT - 1.51e15T^{2} \)
83 \( 1 + (-3.98e7 + 3.98e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 2.38e7iT - 3.93e15T^{2} \)
97 \( 1 + 3.94e7T + 7.83e15T^{2} \)
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

−14.38531550117705140217722633598, −13.50552653494365387011341939451, −12.04104814455200011279775890358, −10.66898020798195017184481771602, −8.675804150701345931062494519719, −8.103413912831956177432208948280, −6.82625190087811087692430588163, −5.29316503131076122130160854272, −3.76123495302572432059088525075, −1.39150559830482130125996680532, 0.938513978020184086559815311547, 2.53080750369622658069736610896, 4.23808130346852845939152549878, 5.21619943397147560040290000640, 8.029252978305958232277507192583, 8.713832465259096239914221025959, 10.39303255651909997667969703595, 11.10117861218262550708285648251, 12.39437221990791730502870837635, 13.55610337804865343797842168878

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