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Properties

Label 2-48-16.11-c4-0-2
Degree $2$
Conductor $48$
Sign $-0.962 - 0.270i$
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  + (0.896 + 3.89i)2-s + (−3.67 − 3.67i)3-s + (−14.3 + 6.98i)4-s + (30.1 + 30.1i)5-s + (11.0 − 17.6i)6-s − 80.6·7-s + (−40.1 − 49.8i)8-s + 27i·9-s + (−90.4 + 144. i)10-s + (−108. + 108. i)11-s + (78.5 + 27.2i)12-s + (11.8 − 11.8i)13-s + (−72.2 − 314. i)14-s − 221. i·15-s + (158. − 201. i)16-s + 67.4·17-s + ⋯
L(s)  = 1  + (0.224 + 0.974i)2-s + (−0.408 − 0.408i)3-s + (−0.899 + 0.436i)4-s + (1.20 + 1.20i)5-s + (0.306 − 0.489i)6-s − 1.64·7-s + (−0.627 − 0.778i)8-s + 0.333i·9-s + (−0.904 + 1.44i)10-s + (−0.893 + 0.893i)11-s + (0.545 + 0.188i)12-s + (0.0700 − 0.0700i)13-s + (−0.368 − 1.60i)14-s − 0.983i·15-s + (0.618 − 0.785i)16-s + 0.233·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.962 - 0.270i$
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: \((2,\ 48,\ (\ :2),\ -0.962 - 0.270i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.144987 + 1.05084i\)
\(L(\frac12)\) \(\approx\) \(0.144987 + 1.05084i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.896 - 3.89i)T \)
3 \( 1 + (3.67 + 3.67i)T \)
good5 \( 1 + (-30.1 - 30.1i)T + 625iT^{2} \)
7 \( 1 + 80.6T + 2.40e3T^{2} \)
11 \( 1 + (108. - 108. i)T - 1.46e4iT^{2} \)
13 \( 1 + (-11.8 + 11.8i)T - 2.85e4iT^{2} \)
17 \( 1 - 67.4T + 8.35e4T^{2} \)
19 \( 1 + (-381. - 381. i)T + 1.30e5iT^{2} \)
23 \( 1 - 284.T + 2.79e5T^{2} \)
29 \( 1 + (-435. + 435. i)T - 7.07e5iT^{2} \)
31 \( 1 - 424. iT - 9.23e5T^{2} \)
37 \( 1 + (404. + 404. i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.39e3iT - 2.82e6T^{2} \)
43 \( 1 + (-1.14e3 + 1.14e3i)T - 3.41e6iT^{2} \)
47 \( 1 - 1.27e3iT - 4.87e6T^{2} \)
53 \( 1 + (1.89e3 + 1.89e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (-2.10e3 + 2.10e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (4.74e3 - 4.74e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (1.37e3 + 1.37e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 993.T + 2.54e7T^{2} \)
73 \( 1 - 5.86e3iT - 2.83e7T^{2} \)
79 \( 1 + 743. iT - 3.89e7T^{2} \)
83 \( 1 + (-6.72e3 - 6.72e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 4.75e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.36e3T + 8.85e7T^{2} \)
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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

−15.47843117577234830157386796013, −14.20893059598124099702412739505, −13.31460645754952079369002354929, −12.46542132341770910894922240805, −10.25937968732021942300930869526, −9.618056893148154668240015891461, −7.41446687170974823654521986878, −6.49770521655666356561948007926, −5.59354468158532983906322513500, −3.01523225666101507861480673518, 0.64019404402054251446899472791, 3.02872472805198524306968600669, 5.04830303024801408448040833955, 6.01171746474995625212036727799, 8.976508408403055388921553746576, 9.624896183587620543868336833951, 10.65386375053685708963761247940, 12.25806892018458113465805667997, 13.18905831852266428822767748795, 13.70068181312687251283921933857

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