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Properties

Label 2-48-16.3-c8-0-15
Degree $2$
Conductor $48$
Sign $-0.252 - 0.967i$
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  + (12.9 + 9.34i)2-s + (−33.0 + 33.0i)3-s + (81.4 + 242. i)4-s + (271. − 271. i)5-s + (−738. + 120. i)6-s + 3.14e3·7-s + (−1.20e3 + 3.91e3i)8-s − 2.18e3i·9-s + (6.05e3 − 989. i)10-s + (1.34e4 + 1.34e4i)11-s + (−1.07e4 − 5.33e3i)12-s + (53.7 + 53.7i)13-s + (4.07e4 + 2.93e4i)14-s + 1.79e4i·15-s + (−5.22e4 + 3.95e4i)16-s − 4.97e4·17-s + ⋯
L(s)  = 1  + (0.811 + 0.583i)2-s + (−0.408 + 0.408i)3-s + (0.318 + 0.948i)4-s + (0.434 − 0.434i)5-s + (−0.569 + 0.0930i)6-s + 1.30·7-s + (−0.295 + 0.955i)8-s − 0.333i·9-s + (0.605 − 0.0989i)10-s + (0.920 + 0.920i)11-s + (−0.516 − 0.257i)12-s + (0.00188 + 0.00188i)13-s + (1.06 + 0.763i)14-s + 0.354i·15-s + (−0.797 + 0.603i)16-s − 0.595·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.252 - 0.967i$
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: \((2,\ 48,\ (\ :4),\ -0.252 - 0.967i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.95085 + 2.52477i\)
\(L(\frac12)\) \(\approx\) \(1.95085 + 2.52477i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-12.9 - 9.34i)T \)
3 \( 1 + (33.0 - 33.0i)T \)
good5 \( 1 + (-271. + 271. i)T - 3.90e5iT^{2} \)
7 \( 1 - 3.14e3T + 5.76e6T^{2} \)
11 \( 1 + (-1.34e4 - 1.34e4i)T + 2.14e8iT^{2} \)
13 \( 1 + (-53.7 - 53.7i)T + 8.15e8iT^{2} \)
17 \( 1 + 4.97e4T + 6.97e9T^{2} \)
19 \( 1 + (8.10e4 - 8.10e4i)T - 1.69e10iT^{2} \)
23 \( 1 - 1.88e4T + 7.83e10T^{2} \)
29 \( 1 + (-1.69e5 - 1.69e5i)T + 5.00e11iT^{2} \)
31 \( 1 + 9.28e5iT - 8.52e11T^{2} \)
37 \( 1 + (-7.66e5 + 7.66e5i)T - 3.51e12iT^{2} \)
41 \( 1 - 5.02e6iT - 7.98e12T^{2} \)
43 \( 1 + (-2.95e6 - 2.95e6i)T + 1.16e13iT^{2} \)
47 \( 1 + 3.96e6iT - 2.38e13T^{2} \)
53 \( 1 + (-8.73e6 + 8.73e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (9.75e6 + 9.75e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (4.29e6 + 4.29e6i)T + 1.91e14iT^{2} \)
67 \( 1 + (-6.69e6 + 6.69e6i)T - 4.06e14iT^{2} \)
71 \( 1 - 2.38e7T + 6.45e14T^{2} \)
73 \( 1 + 4.01e7iT - 8.06e14T^{2} \)
79 \( 1 + 7.02e7iT - 1.51e15T^{2} \)
83 \( 1 + (-7.55e6 + 7.55e6i)T - 2.25e15iT^{2} \)
89 \( 1 - 1.01e8iT - 3.93e15T^{2} \)
97 \( 1 - 2.10e7T + 7.83e15T^{2} \)
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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

−14.55966232356986941559710423022, −13.18657759743744934938795533455, −12.02217419758185621984776549506, −11.07613308495373384547161960692, −9.299948963187161499766376487969, −7.929658515153216935830248811774, −6.40963604084861771585943742022, −5.05152386751601589053916752904, −4.18051439669924710010991644971, −1.81322804036135123967173192665, 1.02940079090269888661673308882, 2.36875036533323896185348065158, 4.31549731862719337358196941058, 5.71533279462857420556298543777, 6.86085749561542396994294499174, 8.774235065143371902688096472958, 10.61497943700700893702756220875, 11.28700594798301803323828734815, 12.30639293150517653029399471027, 13.78248470518467383063439537123

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