Properties

Label 2-48-3.2-c18-0-23
Degree $2$
Conductor $48$
Sign $0.215 + 0.976i$
Analytic cond. $98.5853$
Root an. cond. $9.92901$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.23e3 + 1.92e4i)3-s − 1.14e6i·5-s + 9.81e6·7-s + (−3.51e8 + 1.62e8i)9-s + 2.16e9i·11-s − 1.47e10·13-s + (2.19e10 − 4.83e9i)15-s − 1.36e10i·17-s + 2.38e11·19-s + (4.15e10 + 1.88e11i)21-s − 5.70e11i·23-s + 2.50e12·25-s + (−4.61e12 − 6.06e12i)27-s + 1.35e13i·29-s + 1.17e13·31-s + ⋯
L(s)  = 1  + (0.215 + 0.976i)3-s − 0.584i·5-s + 0.243·7-s + (−0.907 + 0.420i)9-s + 0.917i·11-s − 1.38·13-s + (0.571 − 0.125i)15-s − 0.115i·17-s + 0.740·19-s + (0.0522 + 0.237i)21-s − 0.316i·23-s + 0.657·25-s + (−0.605 − 0.795i)27-s + 0.932i·29-s + 0.442·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.215 + 0.976i$
Analytic conductor: \(98.5853\)
Root analytic conductor: \(9.92901\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :9),\ 0.215 + 0.976i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.8437045028\)
\(L(\frac12)\) \(\approx\) \(0.8437045028\)
\(L(10)\) 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 \( 1 + (-4.23e3 - 1.92e4i)T \)
good5 \( 1 + 1.14e6iT - 3.81e12T^{2} \)
7 \( 1 - 9.81e6T + 1.62e15T^{2} \)
11 \( 1 - 2.16e9iT - 5.55e18T^{2} \)
13 \( 1 + 1.47e10T + 1.12e20T^{2} \)
17 \( 1 + 1.36e10iT - 1.40e22T^{2} \)
19 \( 1 - 2.38e11T + 1.04e23T^{2} \)
23 \( 1 + 5.70e11iT - 3.24e24T^{2} \)
29 \( 1 - 1.35e13iT - 2.10e26T^{2} \)
31 \( 1 - 1.17e13T + 6.99e26T^{2} \)
37 \( 1 + 1.26e14T + 1.68e28T^{2} \)
41 \( 1 + 3.28e14iT - 1.07e29T^{2} \)
43 \( 1 + 8.40e14T + 2.52e29T^{2} \)
47 \( 1 - 9.97e14iT - 1.25e30T^{2} \)
53 \( 1 + 5.45e15iT - 1.08e31T^{2} \)
59 \( 1 + 8.03e14iT - 7.50e31T^{2} \)
61 \( 1 - 1.04e16T + 1.36e32T^{2} \)
67 \( 1 - 1.50e16T + 7.40e32T^{2} \)
71 \( 1 + 4.47e16iT - 2.10e33T^{2} \)
73 \( 1 + 3.92e16T + 3.46e33T^{2} \)
79 \( 1 - 7.20e16T + 1.43e34T^{2} \)
83 \( 1 + 1.02e17iT - 3.49e34T^{2} \)
89 \( 1 + 5.12e17iT - 1.22e35T^{2} \)
97 \( 1 + 7.28e17T + 5.77e35T^{2} \)
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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

−11.69654626343649629724350171876, −10.28832607768179868134197986286, −9.500584269779802260486753879280, −8.403694896316313816483220615049, −7.08839923078443221346251730535, −5.15044717802927144992111238919, −4.69433362349414773277879108868, −3.20120055875016684965873252688, −1.88686714609136446133804576631, −0.18837114781025652421936738359, 1.04171418008527356139289278084, 2.37537340002313546155608578724, 3.30266302234712167870813276781, 5.18316275634013837387994096949, 6.47508732335417893246604704934, 7.44977173794639795529610419571, 8.457702959525411261061261451273, 9.872946276592255976916867795432, 11.29558296525541747326918786654, 12.12724784304273458495578602163

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