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Properties

Label 2-48-16.5-c3-0-4
Degree $2$
Conductor $48$
Sign $0.788 + 0.614i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.24 − 1.72i)2-s + (2.12 − 2.12i)3-s + (2.04 + 7.73i)4-s + (14.6 + 14.6i)5-s + (−8.41 + 1.09i)6-s − 26.8i·7-s + (8.77 − 20.8i)8-s − 8.99i·9-s + (−7.52 − 57.9i)10-s + (23.1 + 23.1i)11-s + (20.7 + 12.0i)12-s + (13.0 − 13.0i)13-s + (−46.4 + 60.2i)14-s + 61.9·15-s + (−55.6 + 31.5i)16-s − 5.45·17-s + ⋯
L(s)  = 1  + (−0.792 − 0.610i)2-s + (0.408 − 0.408i)3-s + (0.255 + 0.966i)4-s + (1.30 + 1.30i)5-s + (−0.572 + 0.0743i)6-s − 1.45i·7-s + (0.387 − 0.921i)8-s − 0.333i·9-s + (−0.237 − 1.83i)10-s + (0.635 + 0.635i)11-s + (0.498 + 0.290i)12-s + (0.278 − 0.278i)13-s + (−0.885 + 1.15i)14-s + 1.06·15-s + (−0.869 + 0.493i)16-s − 0.0778·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.788 + 0.614i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ 0.788 + 0.614i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.20568 - 0.414186i\)
\(L(\frac12)\) \(\approx\) \(1.20568 - 0.414186i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.24 + 1.72i)T \)
3 \( 1 + (-2.12 + 2.12i)T \)
good5 \( 1 + (-14.6 - 14.6i)T + 125iT^{2} \)
7 \( 1 + 26.8iT - 343T^{2} \)
11 \( 1 + (-23.1 - 23.1i)T + 1.33e3iT^{2} \)
13 \( 1 + (-13.0 + 13.0i)T - 2.19e3iT^{2} \)
17 \( 1 + 5.45T + 4.91e3T^{2} \)
19 \( 1 + (-4.68 + 4.68i)T - 6.85e3iT^{2} \)
23 \( 1 + 34.0iT - 1.21e4T^{2} \)
29 \( 1 + (143. - 143. i)T - 2.43e4iT^{2} \)
31 \( 1 + 97.8T + 2.97e4T^{2} \)
37 \( 1 + (268. + 268. i)T + 5.06e4iT^{2} \)
41 \( 1 - 115. iT - 6.89e4T^{2} \)
43 \( 1 + (-73.4 - 73.4i)T + 7.95e4iT^{2} \)
47 \( 1 + 583.T + 1.03e5T^{2} \)
53 \( 1 + (-163. - 163. i)T + 1.48e5iT^{2} \)
59 \( 1 + (45.5 + 45.5i)T + 2.05e5iT^{2} \)
61 \( 1 + (-187. + 187. i)T - 2.26e5iT^{2} \)
67 \( 1 + (223. - 223. i)T - 3.00e5iT^{2} \)
71 \( 1 + 779. iT - 3.57e5T^{2} \)
73 \( 1 + 34.8iT - 3.89e5T^{2} \)
79 \( 1 - 234.T + 4.93e5T^{2} \)
83 \( 1 + (-34.3 + 34.3i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.12e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.33e3T + 9.12e5T^{2} \)
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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

−14.71154308193631228214265636631, −13.80639407176524079677525546884, −12.86812660695116072591558686668, −11.05420993181148112882186720721, −10.28247348411076936661848326793, −9.299291539963330267089311641982, −7.41673339320383708157434826325, −6.66836229406828240947047267616, −3.47151518336491235875506231898, −1.75318747837500017851449748052, 1.84743615602406385314155691345, 5.21523567042884923232709433863, 6.10812247890403782853126200158, 8.508266256313783172881955812721, 9.030489912811576477303901695052, 9.868463872006019767064443290584, 11.68043146734108963061509039687, 13.25188464711271354609377251716, 14.32676780475899997250292967494, 15.53639718936071689383327086205

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