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Properties

Label 2-48-12.11-c21-0-36
Degree $2$
Conductor $48$
Sign $-0.520 + 0.853i$
Analytic cond. $134.149$
Root an. cond. $11.5822$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.02e5 + 2.43e3i)3-s − 1.44e7i·5-s − 1.26e9i·7-s + (1.04e10 + 4.98e8i)9-s + 6.25e10·11-s − 3.73e11·13-s + (3.52e10 − 1.47e12i)15-s − 7.29e12i·17-s + 1.62e12i·19-s + (3.08e12 − 1.29e14i)21-s + 3.19e14·23-s + 2.67e14·25-s + (1.06e15 + 7.63e13i)27-s + 6.11e14i·29-s − 5.24e15i·31-s + ⋯
L(s)  = 1  + (0.999 + 0.0238i)3-s − 0.662i·5-s − 1.69i·7-s + (0.998 + 0.0476i)9-s + 0.727·11-s − 0.752·13-s + (0.0157 − 0.662i)15-s − 0.877i·17-s + 0.0608i·19-s + (0.0403 − 1.69i)21-s + 1.60·23-s + 0.561·25-s + (0.997 + 0.0713i)27-s + 0.270i·29-s − 1.14i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.520 + 0.853i$
Analytic conductor: \(134.149\)
Root analytic conductor: \(11.5822\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :21/2),\ -0.520 + 0.853i)\)

Particular Values

\(L(11)\) \(\approx\) \(3.485564027\)
\(L(\frac12)\) \(\approx\) \(3.485564027\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.02e5 - 2.43e3i)T \)
good5 \( 1 + 1.44e7iT - 4.76e14T^{2} \)
7 \( 1 + 1.26e9iT - 5.58e17T^{2} \)
11 \( 1 - 6.25e10T + 7.40e21T^{2} \)
13 \( 1 + 3.73e11T + 2.47e23T^{2} \)
17 \( 1 + 7.29e12iT - 6.90e25T^{2} \)
19 \( 1 - 1.62e12iT - 7.14e26T^{2} \)
23 \( 1 - 3.19e14T + 3.94e28T^{2} \)
29 \( 1 - 6.11e14iT - 5.13e30T^{2} \)
31 \( 1 + 5.24e15iT - 2.08e31T^{2} \)
37 \( 1 - 1.14e15T + 8.55e32T^{2} \)
41 \( 1 - 5.72e16iT - 7.38e33T^{2} \)
43 \( 1 + 1.59e17iT - 2.00e34T^{2} \)
47 \( 1 - 5.37e17T + 1.30e35T^{2} \)
53 \( 1 + 4.19e16iT - 1.62e36T^{2} \)
59 \( 1 + 5.79e18T + 1.54e37T^{2} \)
61 \( 1 + 7.14e18T + 3.10e37T^{2} \)
67 \( 1 + 1.96e18iT - 2.22e38T^{2} \)
71 \( 1 - 2.21e19T + 7.52e38T^{2} \)
73 \( 1 + 3.72e19T + 1.34e39T^{2} \)
79 \( 1 - 1.09e20iT - 7.08e39T^{2} \)
83 \( 1 + 1.78e20T + 1.99e40T^{2} \)
89 \( 1 - 5.72e20iT - 8.65e40T^{2} \)
97 \( 1 - 8.41e20T + 5.27e41T^{2} \)
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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

−10.92773258677999270699576685430, −9.748194012160412330148436365119, −8.924413461249210077862300757071, −7.56132266339480122712238413436, −6.93511463035000360551542684549, −4.82841532649453974936798165423, −4.04707493258668086615607672219, −2.87238358843534695083388509188, −1.34969498909906827901333126730, −0.61288887650711572553621636040, 1.45690950015018200280140958087, 2.54456934889903572701190244661, 3.22427103913641354459284619117, 4.77264922664637527230708450066, 6.21711007729799689863485538175, 7.32779305074313640239084255195, 8.706807286433832743461740160255, 9.253432386404249842511776608383, 10.62327537505451916092129688256, 12.04258163711742025252052708710

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