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Properties

Label 2-48-16.11-c8-0-21
Degree $2$
Conductor $48$
Sign $0.0265 + 0.999i$
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  + (1.46 − 15.9i)2-s + (33.0 + 33.0i)3-s + (−251. − 46.6i)4-s + (−173. − 173. i)5-s + (575. − 478. i)6-s + 3.86e3·7-s + (−1.11e3 + 3.94e3i)8-s + 2.18e3i·9-s + (−3.01e3 + 2.50e3i)10-s + (−6.95e3 + 6.95e3i)11-s + (−6.78e3 − 9.86e3i)12-s + (2.91e4 − 2.91e4i)13-s + (5.65e3 − 6.16e4i)14-s − 1.14e4i·15-s + (6.11e4 + 2.34e4i)16-s + 7.58e4·17-s + ⋯
L(s)  = 1  + (0.0914 − 0.995i)2-s + (0.408 + 0.408i)3-s + (−0.983 − 0.182i)4-s + (−0.277 − 0.277i)5-s + (0.443 − 0.369i)6-s + 1.61·7-s + (−0.271 + 0.962i)8-s + 0.333i·9-s + (−0.301 + 0.250i)10-s + (−0.475 + 0.475i)11-s + (−0.327 − 0.475i)12-s + (1.01 − 1.01i)13-s + (0.147 − 1.60i)14-s − 0.226i·15-s + (0.933 + 0.357i)16-s + 0.908·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0265 + 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.0265 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.0265 + 0.999i$
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: \((2,\ 48,\ (\ :4),\ 0.0265 + 0.999i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.63770 - 1.59474i\)
\(L(\frac12)\) \(\approx\) \(1.63770 - 1.59474i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.46 + 15.9i)T \)
3 \( 1 + (-33.0 - 33.0i)T \)
good5 \( 1 + (173. + 173. i)T + 3.90e5iT^{2} \)
7 \( 1 - 3.86e3T + 5.76e6T^{2} \)
11 \( 1 + (6.95e3 - 6.95e3i)T - 2.14e8iT^{2} \)
13 \( 1 + (-2.91e4 + 2.91e4i)T - 8.15e8iT^{2} \)
17 \( 1 - 7.58e4T + 6.97e9T^{2} \)
19 \( 1 + (1.64e5 + 1.64e5i)T + 1.69e10iT^{2} \)
23 \( 1 - 2.49e5T + 7.83e10T^{2} \)
29 \( 1 + (-7.68e5 + 7.68e5i)T - 5.00e11iT^{2} \)
31 \( 1 - 3.58e5iT - 8.52e11T^{2} \)
37 \( 1 + (2.21e6 + 2.21e6i)T + 3.51e12iT^{2} \)
41 \( 1 - 2.80e6iT - 7.98e12T^{2} \)
43 \( 1 + (-6.55e5 + 6.55e5i)T - 1.16e13iT^{2} \)
47 \( 1 - 2.55e6iT - 2.38e13T^{2} \)
53 \( 1 + (-1.00e7 - 1.00e7i)T + 6.22e13iT^{2} \)
59 \( 1 + (-6.95e6 + 6.95e6i)T - 1.46e14iT^{2} \)
61 \( 1 + (1.15e6 - 1.15e6i)T - 1.91e14iT^{2} \)
67 \( 1 + (-1.74e7 - 1.74e7i)T + 4.06e14iT^{2} \)
71 \( 1 + 6.78e5T + 6.45e14T^{2} \)
73 \( 1 + 5.75e6iT - 8.06e14T^{2} \)
79 \( 1 + 2.53e6iT - 1.51e15T^{2} \)
83 \( 1 + (-3.98e7 - 3.98e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 2.38e7iT - 3.93e15T^{2} \)
97 \( 1 + 3.94e7T + 7.83e15T^{2} \)
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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

−13.55610337804865343797842168878, −12.39437221990791730502870837635, −11.10117861218262550708285648251, −10.39303255651909997667969703595, −8.713832465259096239914221025959, −8.029252978305958232277507192583, −5.21619943397147560040290000640, −4.23808130346852845939152549878, −2.53080750369622658069736610896, −0.938513978020184086559815311547, 1.39150559830482130125996680532, 3.76123495302572432059088525075, 5.29316503131076122130160854272, 6.82625190087811087692430588163, 8.103413912831956177432208948280, 8.675804150701345931062494519719, 10.66898020798195017184481771602, 12.04104814455200011279775890358, 13.50552653494365387011341939451, 14.38531550117705140217722633598

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