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Properties

Label 2-48-16.11-c8-0-29
Degree $2$
Conductor $48$
Sign $-0.811 - 0.584i$
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  + (9.64 − 12.7i)2-s + (33.0 + 33.0i)3-s + (−69.9 − 246. i)4-s + (−171. − 171. i)5-s + (741. − 103. i)6-s − 1.88e3·7-s + (−3.81e3 − 1.48e3i)8-s + 2.18e3i·9-s + (−3.85e3 + 536. i)10-s + (−2.20e3 + 2.20e3i)11-s + (5.82e3 − 1.04e4i)12-s + (−7.71e3 + 7.71e3i)13-s + (−1.81e4 + 2.40e4i)14-s − 1.13e4i·15-s + (−5.57e4 + 3.44e4i)16-s − 8.96e4·17-s + ⋯
L(s)  = 1  + (0.602 − 0.797i)2-s + (0.408 + 0.408i)3-s + (−0.273 − 0.961i)4-s + (−0.274 − 0.274i)5-s + (0.571 − 0.0796i)6-s − 0.783·7-s + (−0.932 − 0.361i)8-s + 0.333i·9-s + (−0.385 + 0.0536i)10-s + (−0.150 + 0.150i)11-s + (0.281 − 0.504i)12-s + (−0.270 + 0.270i)13-s + (−0.472 + 0.624i)14-s − 0.224i·15-s + (−0.850 + 0.525i)16-s − 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.811 - 0.584i$
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: \((2,\ 48,\ (\ :4),\ -0.811 - 0.584i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.142797 + 0.442397i\)
\(L(\frac12)\) \(\approx\) \(0.142797 + 0.442397i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.64 + 12.7i)T \)
3 \( 1 + (-33.0 - 33.0i)T \)
good5 \( 1 + (171. + 171. i)T + 3.90e5iT^{2} \)
7 \( 1 + 1.88e3T + 5.76e6T^{2} \)
11 \( 1 + (2.20e3 - 2.20e3i)T - 2.14e8iT^{2} \)
13 \( 1 + (7.71e3 - 7.71e3i)T - 8.15e8iT^{2} \)
17 \( 1 + 8.96e4T + 6.97e9T^{2} \)
19 \( 1 + (4.30e4 + 4.30e4i)T + 1.69e10iT^{2} \)
23 \( 1 + 1.35e5T + 7.83e10T^{2} \)
29 \( 1 + (2.67e5 - 2.67e5i)T - 5.00e11iT^{2} \)
31 \( 1 - 1.04e6iT - 8.52e11T^{2} \)
37 \( 1 + (-5.35e5 - 5.35e5i)T + 3.51e12iT^{2} \)
41 \( 1 + 3.57e6iT - 7.98e12T^{2} \)
43 \( 1 + (-1.59e5 + 1.59e5i)T - 1.16e13iT^{2} \)
47 \( 1 + 3.34e6iT - 2.38e13T^{2} \)
53 \( 1 + (2.33e6 + 2.33e6i)T + 6.22e13iT^{2} \)
59 \( 1 + (-1.64e7 + 1.64e7i)T - 1.46e14iT^{2} \)
61 \( 1 + (-4.05e6 + 4.05e6i)T - 1.91e14iT^{2} \)
67 \( 1 + (1.31e7 + 1.31e7i)T + 4.06e14iT^{2} \)
71 \( 1 - 2.65e7T + 6.45e14T^{2} \)
73 \( 1 + 1.89e7iT - 8.06e14T^{2} \)
79 \( 1 + 3.22e7iT - 1.51e15T^{2} \)
83 \( 1 + (8.65e6 + 8.65e6i)T + 2.25e15iT^{2} \)
89 \( 1 - 7.39e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.58e8T + 7.83e15T^{2} \)
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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

−13.15372553667422965906177612813, −12.20243210404917692524442000793, −10.86409715155208612699697426199, −9.768639330937147397060358704162, −8.693108243484503424359154510549, −6.61937113385662680316350971331, −4.89488465492792109494650341285, −3.68485516332535072251458040559, −2.24271225894020177271686876615, −0.12151188672899250397607347694, 2.69582198186851792441751983072, 4.06253922150991124082497467987, 5.93591331614352118807272583316, 7.06215483979534872953671330976, 8.178683857299563531159084556819, 9.524158980199888825237827882224, 11.41761281225860086848366714088, 12.78296849107786818448939910661, 13.41976776637946640970313883152, 14.70131976737722462062632690474

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