↑

Properties

Label 2-48-16.13-c5-0-5
Degree $2$
Conductor $48$
Sign $0.998 - 0.0502i$
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  + (−2.72 − 4.95i)2-s + (−6.36 − 6.36i)3-s + (−17.1 + 27.0i)4-s + (−7.19 + 7.19i)5-s + (−14.1 + 48.9i)6-s + 96.5i·7-s + (180. + 10.9i)8-s + 81i·9-s + (55.2 + 16.0i)10-s + (92.9 − 92.9i)11-s + (280. − 63.2i)12-s + (159. + 159. i)13-s + (478. − 263. i)14-s + 91.5·15-s + (−438. − 925. i)16-s + 1.93e3·17-s + ⋯
L(s)  = 1  + (−0.482 − 0.875i)2-s + (−0.408 − 0.408i)3-s + (−0.534 + 0.845i)4-s + (−0.128 + 0.128i)5-s + (−0.160 + 0.554i)6-s + 0.744i·7-s + (0.998 + 0.0604i)8-s + 0.333i·9-s + (0.174 + 0.0506i)10-s + (0.231 − 0.231i)11-s + (0.563 − 0.126i)12-s + (0.262 + 0.262i)13-s + (0.652 − 0.359i)14-s + 0.105·15-s + (−0.428 − 0.903i)16-s + 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0502i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.998 - 0.0502i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ 0.998 - 0.0502i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.953440 + 0.0239534i\)
\(L(\frac12)\) \(\approx\) \(0.953440 + 0.0239534i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.72 + 4.95i)T \)
3 \( 1 + (6.36 + 6.36i)T \)
good5 \( 1 + (7.19 - 7.19i)T - 3.12e3iT^{2} \)
7 \( 1 - 96.5iT - 1.68e4T^{2} \)
11 \( 1 + (-92.9 + 92.9i)T - 1.61e5iT^{2} \)
13 \( 1 + (-159. - 159. i)T + 3.71e5iT^{2} \)
17 \( 1 - 1.93e3T + 1.41e6T^{2} \)
19 \( 1 + (-162. - 162. i)T + 2.47e6iT^{2} \)
23 \( 1 - 2.83e3iT - 6.43e6T^{2} \)
29 \( 1 + (-3.87e3 - 3.87e3i)T + 2.05e7iT^{2} \)
31 \( 1 + 5.75e3T + 2.86e7T^{2} \)
37 \( 1 + (3.27e3 - 3.27e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 9.37e3iT - 1.15e8T^{2} \)
43 \( 1 + (-492. + 492. i)T - 1.47e8iT^{2} \)
47 \( 1 + 4.99e3T + 2.29e8T^{2} \)
53 \( 1 + (-1.42e4 + 1.42e4i)T - 4.18e8iT^{2} \)
59 \( 1 + (-1.45e4 + 1.45e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (-2.57e4 - 2.57e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (2.84e4 + 2.84e4i)T + 1.35e9iT^{2} \)
71 \( 1 - 1.42e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.16e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.45e4T + 3.07e9T^{2} \)
83 \( 1 + (6.46e4 + 6.46e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 1.06e5iT - 5.58e9T^{2} \)
97 \( 1 + 1.34e5T + 8.58e9T^{2} \)
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

−14.45161338230160298264095988064, −13.13593601373297885443834445433, −12.06454987202314716907705037598, −11.33819865358440841801493429624, −9.959492168639901458532960129951, −8.687836369257363633542750865974, −7.35893891284283480818235669152, −5.45229191124671186216735816016, −3.32873370782796663855897152792, −1.40398548720342526391471534010, 0.71234993656863959236340031796, 4.19870271473069795532083013791, 5.67651799365410402434147894914, 7.08934786661252070619562065757, 8.378662712124366767238224692043, 9.858483692095134923554730402767, 10.66702940587359312910186252389, 12.31967412912010780997830377945, 13.88455868894049016176700561808, 14.79664009920550314916385516732

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