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Properties

Label 2-48-16.5-c5-0-2
Degree $2$
Conductor $48$
Sign $-0.288 - 0.957i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.65 − 0.140i)2-s + (−6.36 + 6.36i)3-s + (31.9 + 1.59i)4-s + (24.8 + 24.8i)5-s + (36.8 − 35.0i)6-s − 146. i·7-s + (−180. − 13.4i)8-s − 81i·9-s + (−137. − 144. i)10-s + (551. + 551. i)11-s + (−213. + 193. i)12-s + (−716. + 716. i)13-s + (−20.5 + 827. i)14-s − 316.·15-s + (1.01e3 + 101. i)16-s − 724.·17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0248i)2-s + (−0.408 + 0.408i)3-s + (0.998 + 0.0497i)4-s + (0.445 + 0.445i)5-s + (0.418 − 0.397i)6-s − 1.12i·7-s + (−0.997 − 0.0745i)8-s − 0.333i·9-s + (−0.434 − 0.456i)10-s + (1.37 + 1.37i)11-s + (−0.428 + 0.387i)12-s + (−1.17 + 1.17i)13-s + (−0.0280 + 1.12i)14-s − 0.363·15-s + (0.995 + 0.0993i)16-s − 0.608·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.288 - 0.957i$
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: \((2,\ 48,\ (\ :5/2),\ -0.288 - 0.957i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.455429 + 0.613215i\)
\(L(\frac12)\) \(\approx\) \(0.455429 + 0.613215i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.65 + 0.140i)T \)
3 \( 1 + (6.36 - 6.36i)T \)
good5 \( 1 + (-24.8 - 24.8i)T + 3.12e3iT^{2} \)
7 \( 1 + 146. iT - 1.68e4T^{2} \)
11 \( 1 + (-551. - 551. i)T + 1.61e5iT^{2} \)
13 \( 1 + (716. - 716. i)T - 3.71e5iT^{2} \)
17 \( 1 + 724.T + 1.41e6T^{2} \)
19 \( 1 + (946. - 946. i)T - 2.47e6iT^{2} \)
23 \( 1 - 4.84e3iT - 6.43e6T^{2} \)
29 \( 1 + (846. - 846. i)T - 2.05e7iT^{2} \)
31 \( 1 - 1.28e3T + 2.86e7T^{2} \)
37 \( 1 + (2.06e3 + 2.06e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 9.39e3iT - 1.15e8T^{2} \)
43 \( 1 + (-9.67e3 - 9.67e3i)T + 1.47e8iT^{2} \)
47 \( 1 - 808.T + 2.29e8T^{2} \)
53 \( 1 + (9.15e3 + 9.15e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (-1.81e4 - 1.81e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (1.43e4 - 1.43e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (-4.25e4 + 4.25e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 4.50e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.72e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.26e4T + 3.07e9T^{2} \)
83 \( 1 + (2.68e4 - 2.68e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 6.47e3iT - 5.58e9T^{2} \)
97 \( 1 + 1.04e3T + 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.07297436642445172927318070580, −14.17801856014466385798325741388, −12.23885766258065183793952017054, −11.21765863452892778752321616519, −9.948932737767091251289854292736, −9.431879788085794154708895362533, −7.31455314278870620704703900307, −6.54282700033745756871231938954, −4.21074508812144306432657694375, −1.74100883768619240031617123939, 0.56986780456831341802141712478, 2.44211855699925725908805163156, 5.56833069604436608548385371538, 6.64232200210520366927174912963, 8.430408415674858200760704525318, 9.178826516159644313373391939829, 10.71867964593703273963184493090, 11.88081633279280182776226127900, 12.76846459828980179886464518905, 14.53696046257928843703090735276

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