↑

Properties

Label 2-48-48.11-c3-0-3
Degree $2$
Conductor $48$
Sign $-0.983 + 0.181i$
Analytic cond. $2.83209$
Root an. cond. $1.68288$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.775 + 2.72i)2-s + (−4.46 − 2.65i)3-s + (−6.79 + 4.21i)4-s + (−5.27 + 5.27i)5-s + (3.75 − 14.2i)6-s − 22.9·7-s + (−16.7 − 15.2i)8-s + (12.9 + 23.7i)9-s + (−18.4 − 10.2i)10-s + (10.6 + 10.6i)11-s + (41.5 − 0.816i)12-s + (12.7 − 12.7i)13-s + (−17.7 − 62.3i)14-s + (37.5 − 9.57i)15-s + (28.3 − 57.3i)16-s + 134. i·17-s + ⋯
L(s)  = 1  + (0.274 + 0.961i)2-s + (−0.859 − 0.510i)3-s + (−0.849 + 0.527i)4-s + (−0.472 + 0.472i)5-s + (0.255 − 0.966i)6-s − 1.23·7-s + (−0.740 − 0.672i)8-s + (0.478 + 0.878i)9-s + (−0.583 − 0.324i)10-s + (0.291 + 0.291i)11-s + (0.999 − 0.0196i)12-s + (0.271 − 0.271i)13-s + (−0.339 − 1.19i)14-s + (0.646 − 0.164i)15-s + (0.443 − 0.896i)16-s + 1.91i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.983 + 0.181i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ -0.983 + 0.181i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0384429 - 0.420165i\)
\(L(\frac12)\) \(\approx\) \(0.0384429 - 0.420165i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.775 - 2.72i)T \)
3 \( 1 + (4.46 + 2.65i)T \)
good5 \( 1 + (5.27 - 5.27i)T - 125iT^{2} \)
7 \( 1 + 22.9T + 343T^{2} \)
11 \( 1 + (-10.6 - 10.6i)T + 1.33e3iT^{2} \)
13 \( 1 + (-12.7 + 12.7i)T - 2.19e3iT^{2} \)
17 \( 1 - 134. iT - 4.91e3T^{2} \)
19 \( 1 + (46.9 + 46.9i)T + 6.85e3iT^{2} \)
23 \( 1 - 93.7iT - 1.21e4T^{2} \)
29 \( 1 + (161. + 161. i)T + 2.43e4iT^{2} \)
31 \( 1 + 120. iT - 2.97e4T^{2} \)
37 \( 1 + (-2.42 - 2.42i)T + 5.06e4iT^{2} \)
41 \( 1 - 253.T + 6.89e4T^{2} \)
43 \( 1 + (135. - 135. i)T - 7.95e4iT^{2} \)
47 \( 1 + 468.T + 1.03e5T^{2} \)
53 \( 1 + (321. - 321. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-119. - 119. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-310. + 310. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-705. - 705. i)T + 3.00e5iT^{2} \)
71 \( 1 - 501. iT - 3.57e5T^{2} \)
73 \( 1 + 641. iT - 3.89e5T^{2} \)
79 \( 1 - 1.23e3iT - 4.93e5T^{2} \)
83 \( 1 + (87.3 - 87.3i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.19e3T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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

−15.74336015414573200289560346095, −14.93935929519019464934843124785, −13.19425436662054380704983212019, −12.75040201061399337674068979488, −11.25351929688802511117731423313, −9.724382537645433108417027691954, −7.936993297661246941723224894070, −6.72470875693949154058439694261, −5.88283592794541594473906122666, −3.87270144835472004142849342288, 0.32794508780461620624162554361, 3.51560002114643712442167634477, 4.91966594444895074667099716887, 6.47022268443750024775205786660, 8.977443709192618583701862427895, 9.923061854541995504393900479901, 11.16034291980303024040644078861, 12.15379013186149081271707804777, 12.95914924407739120418148743350, 14.42272290952494617095958465420

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