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Properties

Label 2-48-16.11-c8-0-17
Degree $2$
Conductor $48$
Sign $-0.375 - 0.926i$
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  + (11.2 + 11.3i)2-s + (33.0 + 33.0i)3-s + (−1.03 + 255. i)4-s + (306. + 306. i)5-s + (−1.50 + 748. i)6-s + 4.33e3·7-s + (−2.91e3 + 2.87e3i)8-s + 2.18e3i·9-s + (−13.9 + 6.93e3i)10-s + (1.83e4 − 1.83e4i)11-s + (−8.49e3 + 8.43e3i)12-s + (−2.44e4 + 2.44e4i)13-s + (4.88e4 + 4.90e4i)14-s + 2.02e4i·15-s + (−6.55e4 − 528. i)16-s − 3.56e4·17-s + ⋯
L(s)  = 1  + (0.705 + 0.708i)2-s + (0.408 + 0.408i)3-s + (−0.00402 + 0.999i)4-s + (0.490 + 0.490i)5-s + (−0.00116 + 0.577i)6-s + 1.80·7-s + (−0.711 + 0.702i)8-s + 0.333i·9-s + (−0.00139 + 0.693i)10-s + (1.25 − 1.25i)11-s + (−0.409 + 0.406i)12-s + (−0.855 + 0.855i)13-s + (1.27 + 1.27i)14-s + 0.400i·15-s + (−0.999 − 0.00805i)16-s − 0.426·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.375 - 0.926i$
Analytic conductor: \(19.5541\)
Root analytic conductor: \(4.42201\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :4),\ -0.375 - 0.926i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.22336 + 3.29866i\)
\(L(\frac12)\) \(\approx\) \(2.22336 + 3.29866i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-11.2 - 11.3i)T \)
3 \( 1 + (-33.0 - 33.0i)T \)
good5 \( 1 + (-306. - 306. i)T + 3.90e5iT^{2} \)
7 \( 1 - 4.33e3T + 5.76e6T^{2} \)
11 \( 1 + (-1.83e4 + 1.83e4i)T - 2.14e8iT^{2} \)
13 \( 1 + (2.44e4 - 2.44e4i)T - 8.15e8iT^{2} \)
17 \( 1 + 3.56e4T + 6.97e9T^{2} \)
19 \( 1 + (-5.51e3 - 5.51e3i)T + 1.69e10iT^{2} \)
23 \( 1 + 9.29e4T + 7.83e10T^{2} \)
29 \( 1 + (3.11e5 - 3.11e5i)T - 5.00e11iT^{2} \)
31 \( 1 + 1.22e5iT - 8.52e11T^{2} \)
37 \( 1 + (1.50e6 + 1.50e6i)T + 3.51e12iT^{2} \)
41 \( 1 - 2.41e6iT - 7.98e12T^{2} \)
43 \( 1 + (9.52e5 - 9.52e5i)T - 1.16e13iT^{2} \)
47 \( 1 + 4.39e6iT - 2.38e13T^{2} \)
53 \( 1 + (-2.72e6 - 2.72e6i)T + 6.22e13iT^{2} \)
59 \( 1 + (-1.41e7 + 1.41e7i)T - 1.46e14iT^{2} \)
61 \( 1 + (-8.54e6 + 8.54e6i)T - 1.91e14iT^{2} \)
67 \( 1 + (2.60e7 + 2.60e7i)T + 4.06e14iT^{2} \)
71 \( 1 - 2.88e7T + 6.45e14T^{2} \)
73 \( 1 + 3.78e7iT - 8.06e14T^{2} \)
79 \( 1 - 7.03e7iT - 1.51e15T^{2} \)
83 \( 1 + (-4.08e7 - 4.08e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 3.44e6iT - 3.93e15T^{2} \)
97 \( 1 + 4.89e7T + 7.83e15T^{2} \)
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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

−14.27682667042982357248051578597, −13.86061916750957272072674900282, −11.89088164529289280426291529621, −11.04631575874076232941696921517, −9.102633171011077073891374982316, −8.078957328666760606791764418605, −6.63229452401782075911717480059, −5.13703891182323160605939588561, −3.90874805093930441563899931607, −2.08865647095594145290241133286, 1.30027810981464302723957951331, 2.13341146914665969741584758064, 4.29947738488688818561518022310, 5.35833505972790503524383007754, 7.23683544463058657150527104444, 8.835252052084880460123950590563, 10.07260053481108558595003836651, 11.57573075345313111918471011471, 12.35709107747745466933730671518, 13.55119860653874869085094748007

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