↑

Properties

Label 2-48-16.5-c5-0-8
Degree $2$
Conductor $48$
Sign $0.0469 - 0.998i$
Analytic cond. $7.69842$
Root an. cond. $2.77460$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (3.53 + 4.41i)2-s + (6.36 − 6.36i)3-s + (−6.97 + 31.2i)4-s + (37.1 + 37.1i)5-s + (50.6 + 5.58i)6-s + 5.83i·7-s + (−162. + 79.6i)8-s − 81i·9-s + (−32.6 + 295. i)10-s + (461. + 461. i)11-s + (154. + 243. i)12-s + (−43.8 + 43.8i)13-s + (−25.7 + 20.6i)14-s + 473.·15-s + (−926. − 435. i)16-s − 365.·17-s + ⋯
L(s)  = 1  + (0.625 + 0.780i)2-s + (0.408 − 0.408i)3-s + (−0.218 + 0.975i)4-s + (0.665 + 0.665i)5-s + (0.573 + 0.0633i)6-s + 0.0449i·7-s + (−0.897 + 0.440i)8-s − 0.333i·9-s + (−0.103 + 0.935i)10-s + (1.14 + 1.14i)11-s + (0.309 + 0.487i)12-s + (−0.0720 + 0.0720i)13-s + (−0.0351 + 0.0281i)14-s + 0.543·15-s + (−0.904 − 0.425i)16-s − 0.307·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0469 - 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0469 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.0469 - 0.998i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ 0.0469 - 0.998i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.96118 + 1.87111i\)
\(L(\frac12)\) \(\approx\) \(1.96118 + 1.87111i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.53 - 4.41i)T \)
3 \( 1 + (-6.36 + 6.36i)T \)
good5 \( 1 + (-37.1 - 37.1i)T + 3.12e3iT^{2} \)
7 \( 1 - 5.83iT - 1.68e4T^{2} \)
11 \( 1 + (-461. - 461. i)T + 1.61e5iT^{2} \)
13 \( 1 + (43.8 - 43.8i)T - 3.71e5iT^{2} \)
17 \( 1 + 365.T + 1.41e6T^{2} \)
19 \( 1 + (322. - 322. i)T - 2.47e6iT^{2} \)
23 \( 1 + 2.75e3iT - 6.43e6T^{2} \)
29 \( 1 + (-4.46e3 + 4.46e3i)T - 2.05e7iT^{2} \)
31 \( 1 - 4.84e3T + 2.86e7T^{2} \)
37 \( 1 + (5.00e3 + 5.00e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.40e4iT - 1.15e8T^{2} \)
43 \( 1 + (-9.38e3 - 9.38e3i)T + 1.47e8iT^{2} \)
47 \( 1 + 2.20e4T + 2.29e8T^{2} \)
53 \( 1 + (2.01e4 + 2.01e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (-6.59e3 - 6.59e3i)T + 7.14e8iT^{2} \)
61 \( 1 + (-4.88e3 + 4.88e3i)T - 8.44e8iT^{2} \)
67 \( 1 + (2.74e4 - 2.74e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 7.93e4iT - 1.80e9T^{2} \)
73 \( 1 - 7.79e4iT - 2.07e9T^{2} \)
79 \( 1 + 7.41e4T + 3.07e9T^{2} \)
83 \( 1 + (-6.69e4 + 6.69e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 6.84e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.86e4T + 8.58e9T^{2} \)
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

−14.55570409653553861147251491708, −14.16999971636869553667127226468, −12.85980457060154611598696893555, −11.85659746475133598575189059059, −9.935715168288201153052900955626, −8.576176755100726479667974529438, −7.04267916980857968967462634340, −6.26510355684618144977079900555, −4.30035150056943724905528949533, −2.41148640191782215751661714411, 1.33669322401149950670709057969, 3.29236060147463270464356988296, 4.82764349528597900264463609470, 6.22049875173849687966956419023, 8.741268119433035266342609771643, 9.565334899309207516615202561437, 10.89465176204694077276012020193, 12.07196301212298055348690230943, 13.44384017999981707928313536324, 13.99728012090148105015229757837

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