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Properties

Label 2-48-48.5-c2-0-2
Degree $2$
Conductor $48$
Sign $-0.100 - 0.994i$
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 + (1.50 + 2.59i)3-s + (2.87 − 2.78i)4-s + (−2.59 + 2.59i)5-s + (−4.73 − 3.68i)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 + (11.5 + 3.26i)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 + (0.501 + 0.865i)3-s + (0.718 − 0.695i)4-s + (−0.519 + 0.519i)5-s + (−0.789 − 0.613i)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 + (0.962 + 0.272i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.100 - 0.994i$
Analytic conductor: \(1.30790\)
Root analytic conductor: \(1.14363\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1),\ -0.100 - 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.554004 + 0.612896i\)
\(L(\frac12)\) \(\approx\) \(0.554004 + 0.612896i\)
\(L(2)\) 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 \( 1 + (-1.50 - 2.59i)T \)
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

−15.73149582416980997472150709810, −14.98017505152095559519302249544, −14.00541694805344741279198950743, −11.64335034298822632947325745885, −10.96225994293376322296400094802, −9.308318267870829896858792307502, −8.787414904598200260853201300792, −7.21948248804691332063442210103, −5.48084971143525896087084392851, −3.05291248250812855849659129831, 1.31108500769649535779557951449, 3.74794901430006816999097777768, 6.86346304702490703878145833346, 7.71748620565989416238611785280, 8.922274826347324103691265181714, 10.17916976291024958985046216061, 11.83463035657341010477097582825, 12.47182843758887253485102505342, 13.84678252132685573273122994722, 15.19306298333596345700029664526

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