↑

Properties

Label 2-48-16.3-c8-0-19
Degree $2$
Conductor $48$
Sign $0.371 + 0.928i$
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  + (−13.2 − 8.92i)2-s + (33.0 − 33.0i)3-s + (96.5 + 237. i)4-s + (646. − 646. i)5-s + (−734. + 143. i)6-s + 1.26e3·7-s + (835. − 4.00e3i)8-s − 2.18e3i·9-s + (−1.43e4 + 2.81e3i)10-s + (1.17e4 + 1.17e4i)11-s + (1.10e4 + 4.64e3i)12-s + (3.13e4 + 3.13e4i)13-s + (−1.67e4 − 1.12e4i)14-s − 4.27e4i·15-s + (−4.69e4 + 4.57e4i)16-s + 8.64e4·17-s + ⋯
L(s)  = 1  + (−0.829 − 0.558i)2-s + (0.408 − 0.408i)3-s + (0.377 + 0.926i)4-s + (1.03 − 1.03i)5-s + (−0.566 + 0.110i)6-s + 0.526·7-s + (0.204 − 0.978i)8-s − 0.333i·9-s + (−1.43 + 0.281i)10-s + (0.800 + 0.800i)11-s + (0.532 + 0.224i)12-s + (1.09 + 1.09i)13-s + (−0.436 − 0.293i)14-s − 0.845i·15-s + (−0.715 + 0.698i)16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.371 + 0.928i$
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: \((2,\ 48,\ (\ :4),\ 0.371 + 0.928i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.71543 - 1.16142i\)
\(L(\frac12)\) \(\approx\) \(1.71543 - 1.16142i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (13.2 + 8.92i)T \)
3 \( 1 + (-33.0 + 33.0i)T \)
good5 \( 1 + (-646. + 646. i)T - 3.90e5iT^{2} \)
7 \( 1 - 1.26e3T + 5.76e6T^{2} \)
11 \( 1 + (-1.17e4 - 1.17e4i)T + 2.14e8iT^{2} \)
13 \( 1 + (-3.13e4 - 3.13e4i)T + 8.15e8iT^{2} \)
17 \( 1 - 8.64e4T + 6.97e9T^{2} \)
19 \( 1 + (9.53e4 - 9.53e4i)T - 1.69e10iT^{2} \)
23 \( 1 - 1.49e5T + 7.83e10T^{2} \)
29 \( 1 + (5.78e5 + 5.78e5i)T + 5.00e11iT^{2} \)
31 \( 1 + 6.97e5iT - 8.52e11T^{2} \)
37 \( 1 + (-2.39e6 + 2.39e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 3.99e6iT - 7.98e12T^{2} \)
43 \( 1 + (1.32e6 + 1.32e6i)T + 1.16e13iT^{2} \)
47 \( 1 + 7.41e5iT - 2.38e13T^{2} \)
53 \( 1 + (2.39e6 - 2.39e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (-1.67e6 - 1.67e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (9.80e6 + 9.80e6i)T + 1.91e14iT^{2} \)
67 \( 1 + (1.33e7 - 1.33e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 1.32e7T + 6.45e14T^{2} \)
73 \( 1 + 3.97e7iT - 8.06e14T^{2} \)
79 \( 1 - 6.78e7iT - 1.51e15T^{2} \)
83 \( 1 + (-1.53e7 + 1.53e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 1.22e8iT - 3.93e15T^{2} \)
97 \( 1 - 4.61e6T + 7.83e15T^{2} \)
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

−13.41697349871939574426456569762, −12.51488810576095390913396503746, −11.38762744088196806573013254417, −9.707534987529172991342536326551, −9.042357107261454430835933481306, −7.88653710644926438828359523025, −6.24143740056036689324689902285, −4.13386498239355429181290256158, −1.90861557389486396866901994904, −1.25070307202921037996779962477, 1.32666317722051407133101036219, 3.04693090711819554191824487641, 5.52882885250374891404804085457, 6.61030314868006257768824515097, 8.160483803291645692460626022173, 9.251513991956696068782733090407, 10.48256203811671420718288127711, 11.10212650050203643454246724097, 13.51684013119339000532740749155, 14.46437961826040606523012991358

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