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Properties

Label 2-48-16.3-c8-0-21
Degree $2$
Conductor $48$
Sign $0.858 + 0.512i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.57 + 14.9i)2-s + (−33.0 + 33.0i)3-s + (−193. + 167. i)4-s + (214. − 214. i)5-s + (−680. − 311. i)6-s − 321.·7-s + (−3.59e3 − 1.97e3i)8-s − 2.18e3i·9-s + (4.41e3 + 2.02e3i)10-s + (−1.34e4 − 1.34e4i)11-s + (872. − 1.19e4i)12-s + (−3.43e3 − 3.43e3i)13-s + (−1.79e3 − 4.81e3i)14-s + 1.41e4i·15-s + (9.53e3 − 6.48e4i)16-s + 6.59e4·17-s + ⋯
L(s)  = 1  + (0.348 + 0.937i)2-s + (−0.408 + 0.408i)3-s + (−0.756 + 0.653i)4-s + (0.343 − 0.343i)5-s + (−0.524 − 0.240i)6-s − 0.133·7-s + (−0.876 − 0.481i)8-s − 0.333i·9-s + (0.441 + 0.202i)10-s + (−0.918 − 0.918i)11-s + (0.0420 − 0.575i)12-s + (−0.120 − 0.120i)13-s + (−0.0466 − 0.125i)14-s + 0.280i·15-s + (0.145 − 0.989i)16-s + 0.789·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.858 + 0.512i$
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: \((2,\ 48,\ (\ :4),\ 0.858 + 0.512i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.977730 - 0.269879i\)
\(L(\frac12)\) \(\approx\) \(0.977730 - 0.269879i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.57 - 14.9i)T \)
3 \( 1 + (33.0 - 33.0i)T \)
good5 \( 1 + (-214. + 214. i)T - 3.90e5iT^{2} \)
7 \( 1 + 321.T + 5.76e6T^{2} \)
11 \( 1 + (1.34e4 + 1.34e4i)T + 2.14e8iT^{2} \)
13 \( 1 + (3.43e3 + 3.43e3i)T + 8.15e8iT^{2} \)
17 \( 1 - 6.59e4T + 6.97e9T^{2} \)
19 \( 1 + (-1.13e5 + 1.13e5i)T - 1.69e10iT^{2} \)
23 \( 1 + 9.91e4T + 7.83e10T^{2} \)
29 \( 1 + (-7.42e4 - 7.42e4i)T + 5.00e11iT^{2} \)
31 \( 1 + 7.83e5iT - 8.52e11T^{2} \)
37 \( 1 + (-3.85e5 + 3.85e5i)T - 3.51e12iT^{2} \)
41 \( 1 + 2.10e6iT - 7.98e12T^{2} \)
43 \( 1 + (2.77e6 + 2.77e6i)T + 1.16e13iT^{2} \)
47 \( 1 + 6.61e6iT - 2.38e13T^{2} \)
53 \( 1 + (-8.73e5 + 8.73e5i)T - 6.22e13iT^{2} \)
59 \( 1 + (1.79e6 + 1.79e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (1.03e7 + 1.03e7i)T + 1.91e14iT^{2} \)
67 \( 1 + (1.57e7 - 1.57e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 4.51e7T + 6.45e14T^{2} \)
73 \( 1 - 1.41e7iT - 8.06e14T^{2} \)
79 \( 1 + 5.93e6iT - 1.51e15T^{2} \)
83 \( 1 + (-4.07e7 + 4.07e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 3.55e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.43e7T + 7.83e15T^{2} \)
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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

−13.82796363289567558426102883697, −12.99468943095949524133001885741, −11.63354595705565013909235571833, −10.04131198428437428723450300441, −8.824499384543550290211053016253, −7.47001791080374414211435798813, −5.85855890616095010352473993703, −5.05070877760083751289851084826, −3.31088717679684516283398290032, −0.35852890636783934789465798617, 1.47787908361949226994152436918, 2.91909666099852481064528449127, 4.81561914064449449397330299814, 6.10287936391393362688304994667, 7.82573394331102845323825838305, 9.743017081626764466944496084152, 10.47516604005014907305942073249, 11.87853753691014539028196314381, 12.66622783745273060761627243639, 13.79195763557822355058974419916

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