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Properties

Label 2-48-16.11-c8-0-5
Degree $2$
Conductor $48$
Sign $-0.632 - 0.774i$
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  + (1.16 + 15.9i)2-s + (−33.0 − 33.0i)3-s + (−253. + 37.2i)4-s + (−679. − 679. i)5-s + (489. − 566. i)6-s + 4.52e3·7-s + (−891. − 3.99e3i)8-s + 2.18e3i·9-s + (1.00e4 − 1.16e4i)10-s + (−6.53e3 + 6.53e3i)11-s + (9.60e3 + 7.14e3i)12-s + (−1.39e4 + 1.39e4i)13-s + (5.29e3 + 7.22e4i)14-s + 4.49e4i·15-s + (6.27e4 − 1.88e4i)16-s − 1.34e5·17-s + ⋯
L(s)  = 1  + (0.0730 + 0.997i)2-s + (−0.408 − 0.408i)3-s + (−0.989 + 0.145i)4-s + (−1.08 − 1.08i)5-s + (0.377 − 0.436i)6-s + 1.88·7-s + (−0.217 − 0.976i)8-s + 0.333i·9-s + (1.00 − 1.16i)10-s + (−0.446 + 0.446i)11-s + (0.463 + 0.344i)12-s + (−0.489 + 0.489i)13-s + (0.137 + 1.88i)14-s + 0.887i·15-s + (0.957 − 0.288i)16-s − 1.61·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.632 - 0.774i$
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: \((2,\ 48,\ (\ :4),\ -0.632 - 0.774i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.358861 + 0.756663i\)
\(L(\frac12)\) \(\approx\) \(0.358861 + 0.756663i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 - 15.9i)T \)
3 \( 1 + (33.0 + 33.0i)T \)
good5 \( 1 + (679. + 679. i)T + 3.90e5iT^{2} \)
7 \( 1 - 4.52e3T + 5.76e6T^{2} \)
11 \( 1 + (6.53e3 - 6.53e3i)T - 2.14e8iT^{2} \)
13 \( 1 + (1.39e4 - 1.39e4i)T - 8.15e8iT^{2} \)
17 \( 1 + 1.34e5T + 6.97e9T^{2} \)
19 \( 1 + (-3.59e4 - 3.59e4i)T + 1.69e10iT^{2} \)
23 \( 1 - 3.99e5T + 7.83e10T^{2} \)
29 \( 1 + (1.31e5 - 1.31e5i)T - 5.00e11iT^{2} \)
31 \( 1 - 1.67e6iT - 8.52e11T^{2} \)
37 \( 1 + (-5.93e5 - 5.93e5i)T + 3.51e12iT^{2} \)
41 \( 1 - 2.25e6iT - 7.98e12T^{2} \)
43 \( 1 + (3.00e6 - 3.00e6i)T - 1.16e13iT^{2} \)
47 \( 1 - 3.59e6iT - 2.38e13T^{2} \)
53 \( 1 + (-2.22e6 - 2.22e6i)T + 6.22e13iT^{2} \)
59 \( 1 + (-7.26e6 + 7.26e6i)T - 1.46e14iT^{2} \)
61 \( 1 + (-1.25e5 + 1.25e5i)T - 1.91e14iT^{2} \)
67 \( 1 + (1.15e7 + 1.15e7i)T + 4.06e14iT^{2} \)
71 \( 1 + 1.29e7T + 6.45e14T^{2} \)
73 \( 1 - 1.57e7iT - 8.06e14T^{2} \)
79 \( 1 + 4.27e7iT - 1.51e15T^{2} \)
83 \( 1 + (-3.27e7 - 3.27e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 2.14e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.40e7T + 7.83e15T^{2} \)
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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

−14.58559599912684586217538827872, −13.20767674663081533066249204992, −12.14386439228719103927097755912, −11.11656602661369500081144961990, −8.866059572735664170850163805551, −8.055409838683517696951218285237, −7.08655737686628001154573908466, −4.89973619610544014283512671797, −4.67207500238764684383115892797, −1.24893594650293781276369397342, 0.36319874237093456610887191104, 2.45652386344830568852175028528, 4.06243074379524944368119451450, 5.15939872321599520176776378198, 7.48113125148270950926074226532, 8.677147147655691889817535251761, 10.56985723010358506473177974240, 11.21613868108404247320367354306, 11.69016892801707542300212615077, 13.44034866937020691253447435745

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