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Properties

Label 2-48-48.29-c2-0-4
Degree $2$
Conductor $48$
Sign $0.581 - 0.813i$
Analytic cond. $1.30790$
Root an. cond. $1.14363$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.85 + 0.750i)2-s + (−2.59 + 1.50i)3-s + (2.87 + 2.78i)4-s + (2.59 + 2.59i)5-s + (−5.94 + 0.841i)6-s − 7.30i·7-s + (3.23 + 7.31i)8-s + (4.47 − 7.81i)9-s + (2.86 + 6.76i)10-s + (−11.3 − 11.3i)11-s + (−11.6 − 2.89i)12-s + (−0.746 − 0.746i)13-s + (5.47 − 13.5i)14-s + (−10.6 − 2.83i)15-s + (0.510 + 15.9i)16-s − 6.67i·17-s + ⋯
L(s)  = 1  + (0.926 + 0.375i)2-s + (−0.865 + 0.501i)3-s + (0.718 + 0.695i)4-s + (0.519 + 0.519i)5-s + (−0.990 + 0.140i)6-s − 1.04i·7-s + (0.404 + 0.914i)8-s + (0.496 − 0.867i)9-s + (0.286 + 0.676i)10-s + (−1.02 − 1.02i)11-s + (−0.970 − 0.241i)12-s + (−0.0574 − 0.0574i)13-s + (0.391 − 0.966i)14-s + (−0.710 − 0.188i)15-s + (0.0319 + 0.999i)16-s − 0.392i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(1.30790\)
Root analytic conductor: \(1.14363\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1),\ 0.581 - 0.813i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.33299 + 0.685626i\)
\(L(\frac12)\) \(\approx\) \(1.33299 + 0.685626i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.85 - 0.750i)T \)
3 \( 1 + (2.59 - 1.50i)T \)
good5 \( 1 + (-2.59 - 2.59i)T + 25iT^{2} \)
7 \( 1 + 7.30iT - 49T^{2} \)
11 \( 1 + (11.3 + 11.3i)T + 121iT^{2} \)
13 \( 1 + (0.746 + 0.746i)T + 169iT^{2} \)
17 \( 1 + 6.67iT - 289T^{2} \)
19 \( 1 + (-22.1 - 22.1i)T + 361iT^{2} \)
23 \( 1 + 21.4T + 529T^{2} \)
29 \( 1 + (1.54 - 1.54i)T - 841iT^{2} \)
31 \( 1 + 14.6T + 961T^{2} \)
37 \( 1 + (50.1 - 50.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 15.0T + 1.68e3T^{2} \)
43 \( 1 + (-26.3 + 26.3i)T - 1.84e3iT^{2} \)
47 \( 1 - 36.6iT - 2.20e3T^{2} \)
53 \( 1 + (50.9 + 50.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (-12.1 - 12.1i)T + 3.48e3iT^{2} \)
61 \( 1 + (27.5 + 27.5i)T + 3.72e3iT^{2} \)
67 \( 1 + (4.84 + 4.84i)T + 4.48e3iT^{2} \)
71 \( 1 - 74.9T + 5.04e3T^{2} \)
73 \( 1 - 3.47iT - 5.32e3T^{2} \)
79 \( 1 - 103.T + 6.24e3T^{2} \)
83 \( 1 + (-31.7 + 31.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 78.2T + 7.92e3T^{2} \)
97 \( 1 + 61.5T + 9.40e3T^{2} \)
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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

−15.81986659411752424962672772099, −14.26619767264856873316716899186, −13.58680647786923032219724111955, −12.16324673990203501147921307781, −10.92660360937330014401763516955, −10.13761538572382035101678500857, −7.73348889867091050487244673092, −6.33887549095392868411410366912, −5.23409819814369837278429855110, −3.53246933964722492058647193015, 2.11989836425608620263998405131, 4.98822413438606847223743469945, 5.74755044269357886450030307545, 7.33926731964791750827191611163, 9.544888977799611184574659750424, 10.87763874354995399153250929195, 12.14316007843681744260334010036, 12.73475449415733630376487795580, 13.73851880110252909084218814661, 15.33926862086716605355589180827

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