Properties

Label 2-48-48.11-c3-0-16
Degree $2$
Conductor $48$
Sign $0.987 + 0.159i$
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  + (2.56 + 1.18i)2-s + (−1.96 − 4.81i)3-s + (5.20 + 6.07i)4-s + (6.30 − 6.30i)5-s + (0.652 − 14.6i)6-s + 24.6·7-s + (6.17 + 21.7i)8-s + (−19.3 + 18.8i)9-s + (23.6 − 8.73i)10-s + (−40.4 − 40.4i)11-s + (19.0 − 36.9i)12-s + (−47.3 + 47.3i)13-s + (63.3 + 29.1i)14-s + (−42.6 − 17.9i)15-s + (−9.86 + 63.2i)16-s + 41.7i·17-s + ⋯
L(s)  = 1  + (0.908 + 0.418i)2-s + (−0.377 − 0.926i)3-s + (0.650 + 0.759i)4-s + (0.563 − 0.563i)5-s + (0.0444 − 0.999i)6-s + 1.33·7-s + (0.273 + 0.961i)8-s + (−0.715 + 0.698i)9-s + (0.747 − 0.276i)10-s + (−1.10 − 1.10i)11-s + (0.458 − 0.888i)12-s + (−1.00 + 1.00i)13-s + (1.21 + 0.557i)14-s + (−0.734 − 0.309i)15-s + (−0.154 + 0.988i)16-s + 0.595i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.987 + 0.159i$
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: \((2,\ 48,\ (\ :3/2),\ 0.987 + 0.159i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.12835 - 0.171170i\)
\(L(\frac12)\) \(\approx\) \(2.12835 - 0.171170i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.56 - 1.18i)T \)
3 \( 1 + (1.96 + 4.81i)T \)
good5 \( 1 + (-6.30 + 6.30i)T - 125iT^{2} \)
7 \( 1 - 24.6T + 343T^{2} \)
11 \( 1 + (40.4 + 40.4i)T + 1.33e3iT^{2} \)
13 \( 1 + (47.3 - 47.3i)T - 2.19e3iT^{2} \)
17 \( 1 - 41.7iT - 4.91e3T^{2} \)
19 \( 1 + (-10.6 - 10.6i)T + 6.85e3iT^{2} \)
23 \( 1 + 53.4iT - 1.21e4T^{2} \)
29 \( 1 + (105. + 105. i)T + 2.43e4iT^{2} \)
31 \( 1 + 3.14iT - 2.97e4T^{2} \)
37 \( 1 + (-42.1 - 42.1i)T + 5.06e4iT^{2} \)
41 \( 1 - 152.T + 6.89e4T^{2} \)
43 \( 1 + (-221. + 221. i)T - 7.95e4iT^{2} \)
47 \( 1 + 381.T + 1.03e5T^{2} \)
53 \( 1 + (-294. + 294. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-445. - 445. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-21.8 + 21.8i)T - 2.26e5iT^{2} \)
67 \( 1 + (572. + 572. i)T + 3.00e5iT^{2} \)
71 \( 1 + 612. iT - 3.57e5T^{2} \)
73 \( 1 + 331. iT - 3.89e5T^{2} \)
79 \( 1 - 427. iT - 4.93e5T^{2} \)
83 \( 1 + (245. - 245. i)T - 5.71e5iT^{2} \)
89 \( 1 + 188.T + 7.04e5T^{2} \)
97 \( 1 - 1.47e3T + 9.12e5T^{2} \)
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

−14.78650895647815606940420588901, −13.84685556054834191562790806732, −13.04281904603456380191335759600, −11.88370367096886009702555702096, −10.95648631821923044705693540677, −8.474973114667108107346408981679, −7.49420232946825100731005182139, −5.85713477028461184189598107632, −4.90483857406332702297073488478, −2.08419165028688363260848533022, 2.58440954758939185868056259473, 4.71935356857537284637511409056, 5.49022713111497505408612847917, 7.45847405679327846514072446446, 9.811663317039717477332022794417, 10.56036306006949372768424063941, 11.56471541533701423286312604240, 12.83521248274354101173664580129, 14.40897014646717199058231808322, 14.88083692263812289720359516460

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