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Properties

Label 2-48-16.11-c4-0-11
Degree $2$
Conductor $48$
Sign $0.960 + 0.278i$
Analytic cond. $4.96175$
Root an. cond. $2.22750$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.90 − 0.888i)2-s + (3.67 + 3.67i)3-s + (14.4 − 6.92i)4-s + (−17.3 − 17.3i)5-s + (17.5 + 11.0i)6-s + 89.8·7-s + (50.0 − 39.8i)8-s + 27i·9-s + (−83.1 − 52.3i)10-s + (−94.2 + 94.2i)11-s + (78.4 + 27.5i)12-s + (−100. + 100. i)13-s + (350. − 79.7i)14-s − 127. i·15-s + (160. − 199. i)16-s − 66.1·17-s + ⋯
L(s)  = 1  + (0.975 − 0.222i)2-s + (0.408 + 0.408i)3-s + (0.901 − 0.432i)4-s + (−0.694 − 0.694i)5-s + (0.488 + 0.307i)6-s + 1.83·7-s + (0.782 − 0.622i)8-s + 0.333i·9-s + (−0.831 − 0.523i)10-s + (−0.779 + 0.779i)11-s + (0.544 + 0.191i)12-s + (−0.596 + 0.596i)13-s + (1.78 − 0.407i)14-s − 0.567i·15-s + (0.625 − 0.780i)16-s − 0.228·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.960 + 0.278i$
Analytic conductor: \(4.96175\)
Root analytic conductor: \(2.22750\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :2),\ 0.960 + 0.278i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.89745 - 0.412045i\)
\(L(\frac12)\) \(\approx\) \(2.89745 - 0.412045i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.90 + 0.888i)T \)
3 \( 1 + (-3.67 - 3.67i)T \)
good5 \( 1 + (17.3 + 17.3i)T + 625iT^{2} \)
7 \( 1 - 89.8T + 2.40e3T^{2} \)
11 \( 1 + (94.2 - 94.2i)T - 1.46e4iT^{2} \)
13 \( 1 + (100. - 100. i)T - 2.85e4iT^{2} \)
17 \( 1 + 66.1T + 8.35e4T^{2} \)
19 \( 1 + (324. + 324. i)T + 1.30e5iT^{2} \)
23 \( 1 + 467.T + 2.79e5T^{2} \)
29 \( 1 + (321. - 321. i)T - 7.07e5iT^{2} \)
31 \( 1 - 1.58e3iT - 9.23e5T^{2} \)
37 \( 1 + (-982. - 982. i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.21e3iT - 2.82e6T^{2} \)
43 \( 1 + (-1.30e3 + 1.30e3i)T - 3.41e6iT^{2} \)
47 \( 1 + 3.97e3iT - 4.87e6T^{2} \)
53 \( 1 + (-672. - 672. i)T + 7.89e6iT^{2} \)
59 \( 1 + (1.18e3 - 1.18e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (979. - 979. i)T - 1.38e7iT^{2} \)
67 \( 1 + (2.26e3 + 2.26e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 5.87e3T + 2.54e7T^{2} \)
73 \( 1 + 3.66e3iT - 2.83e7T^{2} \)
79 \( 1 - 539. iT - 3.89e7T^{2} \)
83 \( 1 + (3.60e3 + 3.60e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 7.54e3iT - 6.27e7T^{2} \)
97 \( 1 + 9.59e3T + 8.85e7T^{2} \)
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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

−14.86306879984909246133420360023, −13.86563931598327716553801599793, −12.47737207383960717693781866765, −11.54141613578511619557287965978, −10.43622099395609515708893432739, −8.541079690013912726685138264219, −7.39290151720046191796674707182, −4.93199223812630055316330831767, −4.42800438555064968161042700815, −2.05914216131395043421815910228, 2.35528311257401121496858027811, 4.14051611204220703200694136361, 5.77724068340738282159196527374, 7.74465208087770285652743420846, 7.962150603521547065725421800790, 10.80318888548558341200684613161, 11.50334146848028200388130747796, 12.77102068555833428049933948028, 14.10440419691923239329087748771, 14.75034477865395554324242168921

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