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Properties

Label 2-48-16.11-c8-0-0
Degree $2$
Conductor $48$
Sign $-0.606 + 0.794i$
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  + (8.00 + 13.8i)2-s + (−33.0 − 33.0i)3-s + (−127. + 221. i)4-s + (318. + 318. i)5-s + (193. − 722. i)6-s − 621.·7-s + (−4.09e3 + 7.29i)8-s + 2.18e3i·9-s + (−1.86e3 + 6.96e3i)10-s + (−4.41e3 + 4.41e3i)11-s + (1.15e4 − 3.11e3i)12-s + (−772. + 772. i)13-s + (−4.97e3 − 8.60e3i)14-s − 2.10e4i·15-s + (−3.29e4 − 5.66e4i)16-s − 3.98e4·17-s + ⋯
L(s)  = 1  + (0.500 + 0.865i)2-s + (−0.408 − 0.408i)3-s + (−0.498 + 0.866i)4-s + (0.510 + 0.510i)5-s + (0.149 − 0.557i)6-s − 0.258·7-s + (−0.999 + 0.00178i)8-s + 0.333i·9-s + (−0.186 + 0.696i)10-s + (−0.301 + 0.301i)11-s + (0.557 − 0.150i)12-s + (−0.0270 + 0.0270i)13-s + (−0.129 − 0.224i)14-s − 0.416i·15-s + (−0.502 − 0.864i)16-s − 0.476·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.794i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.606 + 0.794i$
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: \((2,\ 48,\ (\ :4),\ -0.606 + 0.794i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.144326 - 0.291791i\)
\(L(\frac12)\) \(\approx\) \(0.144326 - 0.291791i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-8.00 - 13.8i)T \)
3 \( 1 + (33.0 + 33.0i)T \)
good5 \( 1 + (-318. - 318. i)T + 3.90e5iT^{2} \)
7 \( 1 + 621.T + 5.76e6T^{2} \)
11 \( 1 + (4.41e3 - 4.41e3i)T - 2.14e8iT^{2} \)
13 \( 1 + (772. - 772. i)T - 8.15e8iT^{2} \)
17 \( 1 + 3.98e4T + 6.97e9T^{2} \)
19 \( 1 + (1.34e5 + 1.34e5i)T + 1.69e10iT^{2} \)
23 \( 1 + 2.87e5T + 7.83e10T^{2} \)
29 \( 1 + (2.06e4 - 2.06e4i)T - 5.00e11iT^{2} \)
31 \( 1 + 8.26e5iT - 8.52e11T^{2} \)
37 \( 1 + (-4.63e5 - 4.63e5i)T + 3.51e12iT^{2} \)
41 \( 1 - 2.66e6iT - 7.98e12T^{2} \)
43 \( 1 + (3.62e6 - 3.62e6i)T - 1.16e13iT^{2} \)
47 \( 1 - 8.55e5iT - 2.38e13T^{2} \)
53 \( 1 + (-3.79e6 - 3.79e6i)T + 6.22e13iT^{2} \)
59 \( 1 + (2.22e6 - 2.22e6i)T - 1.46e14iT^{2} \)
61 \( 1 + (3.05e6 - 3.05e6i)T - 1.91e14iT^{2} \)
67 \( 1 + (-7.26e6 - 7.26e6i)T + 4.06e14iT^{2} \)
71 \( 1 - 1.90e7T + 6.45e14T^{2} \)
73 \( 1 - 4.82e7iT - 8.06e14T^{2} \)
79 \( 1 - 3.72e7iT - 1.51e15T^{2} \)
83 \( 1 + (5.70e7 + 5.70e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 2.95e7iT - 3.93e15T^{2} \)
97 \( 1 - 2.58e7T + 7.83e15T^{2} \)
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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

−14.65860290907063636482931893981, −13.50872546100595817719870710052, −12.74323347245849198668979487622, −11.36596247878876241771804647872, −9.820308034988270873558011289970, −8.234887429534279819106348216244, −6.84911098315458615244994397982, −6.04209684922340149160387718711, −4.50494062772823402874330250184, −2.52162845826629365535447905699, 0.098787419784861415370472131505, 1.86386602068636178370134741452, 3.71239283289160598984400228973, 5.11616615430218884612125807621, 6.21422702827506490671264660233, 8.672212914309573008835468344792, 9.889017432663160422571512733349, 10.77455781225430870102657510783, 12.08877776488344009718859671768, 13.00596562276532977328058121230

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