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Properties

Label 2-48-16.11-c6-0-3
Degree $2$
Conductor $48$
Sign $-0.326 - 0.945i$
Analytic cond. $11.0425$
Root an. cond. $3.32304$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.74 + 5.57i)2-s + (−11.0 − 11.0i)3-s + (1.92 − 63.9i)4-s + (−160. − 160. i)5-s + (124. + 1.87i)6-s − 53.5·7-s + (345. + 377. i)8-s + 242. i·9-s + (1.81e3 + 27.2i)10-s + (−122. + 122. i)11-s + (−726. + 683. i)12-s + (413. − 413. i)13-s + (307. − 298. i)14-s + 3.53e3i·15-s + (−4.08e3 − 245. i)16-s − 3.29e3·17-s + ⋯
L(s)  = 1  + (−0.717 + 0.696i)2-s + (−0.408 − 0.408i)3-s + (0.0300 − 0.999i)4-s + (−1.28 − 1.28i)5-s + (0.577 + 0.00866i)6-s − 0.156·7-s + (0.674 + 0.738i)8-s + 0.333i·9-s + (1.81 + 0.0272i)10-s + (−0.0919 + 0.0919i)11-s + (−0.420 + 0.395i)12-s + (0.188 − 0.188i)13-s + (0.111 − 0.108i)14-s + 1.04i·15-s + (−0.998 − 0.0599i)16-s − 0.670·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.326 - 0.945i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ -0.326 - 0.945i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.151280 + 0.212325i\)
\(L(\frac12)\) \(\approx\) \(0.151280 + 0.212325i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.74 - 5.57i)T \)
3 \( 1 + (11.0 + 11.0i)T \)
good5 \( 1 + (160. + 160. i)T + 1.56e4iT^{2} \)
7 \( 1 + 53.5T + 1.17e5T^{2} \)
11 \( 1 + (122. - 122. i)T - 1.77e6iT^{2} \)
13 \( 1 + (-413. + 413. i)T - 4.82e6iT^{2} \)
17 \( 1 + 3.29e3T + 2.41e7T^{2} \)
19 \( 1 + (-7.93e3 - 7.93e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 1.36e4T + 1.48e8T^{2} \)
29 \( 1 + (-5.98e3 + 5.98e3i)T - 5.94e8iT^{2} \)
31 \( 1 + 2.31e4iT - 8.87e8T^{2} \)
37 \( 1 + (-6.29e4 - 6.29e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 9.07e4iT - 4.75e9T^{2} \)
43 \( 1 + (-1.01e5 + 1.01e5i)T - 6.32e9iT^{2} \)
47 \( 1 - 3.26e4iT - 1.07e10T^{2} \)
53 \( 1 + (2.86e4 + 2.86e4i)T + 2.21e10iT^{2} \)
59 \( 1 + (1.35e5 - 1.35e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (2.47e5 - 2.47e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (5.23e4 + 5.23e4i)T + 9.04e10iT^{2} \)
71 \( 1 - 3.81e5T + 1.28e11T^{2} \)
73 \( 1 - 1.24e5iT - 1.51e11T^{2} \)
79 \( 1 + 2.23e5iT - 2.43e11T^{2} \)
83 \( 1 + (5.68e5 + 5.68e5i)T + 3.26e11iT^{2} \)
89 \( 1 - 8.63e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.32e6T + 8.32e11T^{2} \)
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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

−15.22591476125432884396001612784, −13.59180762308157227920566036734, −12.27983590518771101232354988380, −11.32881965329629435475996068114, −9.678194388762790772707162005238, −8.285805076650796951170256661248, −7.60725063898957347882755700536, −5.89986845748733395869061156306, −4.46125100785970431557805512862, −1.12109813469608964715339483474, 0.19015709343838339049924901595, 2.90925325861863821215655563880, 4.14588219222223553456912825531, 6.76505176078335035406719716978, 7.85634993078354637629595279761, 9.420512952607852356796454014121, 10.82146210763903488213011794349, 11.28374844127693434826023640868, 12.36073061627770460614604159074, 14.09338980544743473041565203679

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