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Properties

Label 2-48-48.5-c4-0-8
Degree $2$
Conductor $48$
Sign $-0.994 + 0.102i$
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  + (0.312 + 3.98i)2-s + (1.63 + 8.85i)3-s + (−15.8 + 2.49i)4-s + (−2.85 + 2.85i)5-s + (−34.7 + 9.28i)6-s + 27.2i·7-s + (−14.8 − 62.2i)8-s + (−75.6 + 28.9i)9-s + (−12.2 − 10.5i)10-s + (77.0 − 77.0i)11-s + (−47.8 − 135. i)12-s + (−63.4 + 63.4i)13-s + (−108. + 8.51i)14-s + (−29.9 − 20.6i)15-s + (243. − 78.7i)16-s + 161. i·17-s + ⋯
L(s)  = 1  + (0.0781 + 0.996i)2-s + (0.181 + 0.983i)3-s + (−0.987 + 0.155i)4-s + (−0.114 + 0.114i)5-s + (−0.966 + 0.257i)6-s + 0.556i·7-s + (−0.232 − 0.972i)8-s + (−0.933 + 0.357i)9-s + (−0.122 − 0.105i)10-s + (0.636 − 0.636i)11-s + (−0.332 − 0.943i)12-s + (−0.375 + 0.375i)13-s + (−0.554 + 0.0434i)14-s + (−0.133 − 0.0916i)15-s + (0.951 − 0.307i)16-s + 0.560i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.994 + 0.102i$
Analytic conductor: \(4.96175\)
Root analytic conductor: \(2.22750\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :2),\ -0.994 + 0.102i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0635072 - 1.23495i\)
\(L(\frac12)\) \(\approx\) \(0.0635072 - 1.23495i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.312 - 3.98i)T \)
3 \( 1 + (-1.63 - 8.85i)T \)
good5 \( 1 + (2.85 - 2.85i)T - 625iT^{2} \)
7 \( 1 - 27.2iT - 2.40e3T^{2} \)
11 \( 1 + (-77.0 + 77.0i)T - 1.46e4iT^{2} \)
13 \( 1 + (63.4 - 63.4i)T - 2.85e4iT^{2} \)
17 \( 1 - 161. iT - 8.35e4T^{2} \)
19 \( 1 + (390. - 390. i)T - 1.30e5iT^{2} \)
23 \( 1 - 720.T + 2.79e5T^{2} \)
29 \( 1 + (-693. - 693. i)T + 7.07e5iT^{2} \)
31 \( 1 - 786.T + 9.23e5T^{2} \)
37 \( 1 + (-538. - 538. i)T + 1.87e6iT^{2} \)
41 \( 1 + 925.T + 2.82e6T^{2} \)
43 \( 1 + (1.59e3 + 1.59e3i)T + 3.41e6iT^{2} \)
47 \( 1 + 3.18e3iT - 4.87e6T^{2} \)
53 \( 1 + (152. - 152. i)T - 7.89e6iT^{2} \)
59 \( 1 + (-465. + 465. i)T - 1.21e7iT^{2} \)
61 \( 1 + (3.24e3 - 3.24e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (-2.95e3 + 2.95e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 6.53e3T + 2.54e7T^{2} \)
73 \( 1 + 9.36e3iT - 2.83e7T^{2} \)
79 \( 1 - 7.59e3T + 3.89e7T^{2} \)
83 \( 1 + (-4.13e3 - 4.13e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 3.26e3T + 6.27e7T^{2} \)
97 \( 1 + 1.05e4T + 8.85e7T^{2} \)
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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

−15.25394607028744595481897368284, −14.81221736160008864350170556514, −13.64209605104930421636796485026, −12.08540403094262752457957116699, −10.51026840326289187218780655079, −9.142769814374931662483243790277, −8.345033295372582346420905396960, −6.47837577597311333217498223148, −5.10147866237209482616982473623, −3.60783904887297711414610177375, 0.77150873857542913194436434777, 2.62575186229691777188715727888, 4.60102930497993366446597427365, 6.71531576673412339910738024824, 8.231350825194034916441682387004, 9.520408361935010298011211123599, 10.99541820280285880842456541416, 12.10280469508565209842613242271, 12.99611487833595723925277851222, 13.94119055664549980628992289218

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