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Properties

Label 2-48-12.11-c21-0-10
Degree $2$
Conductor $48$
Sign $0.469 - 0.882i$
Analytic cond. $134.149$
Root an. cond. $11.5822$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.02e5 + 3.56e3i)3-s − 3.58e7i·5-s + 2.03e8i·7-s + (1.04e10 − 7.28e8i)9-s + 1.20e11·11-s + 1.77e11·13-s + (1.27e11 + 3.66e12i)15-s + 1.07e13i·17-s − 7.48e12i·19-s + (−7.24e11 − 2.07e13i)21-s + 2.50e14·23-s − 8.11e14·25-s + (−1.06e15 + 1.11e14i)27-s − 1.36e15i·29-s + 7.10e15i·31-s + ⋯
L(s)  = 1  + (−0.999 + 0.0348i)3-s − 1.64i·5-s + 0.272i·7-s + (0.997 − 0.0696i)9-s + 1.40·11-s + 0.356·13-s + (0.0572 + 1.64i)15-s + 1.29i·17-s − 0.280i·19-s + (−0.00948 − 0.272i)21-s + 1.26·23-s − 1.70·25-s + (−0.994 + 0.104i)27-s − 0.604i·29-s + 1.55i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.882i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.469 - 0.882i$
Analytic conductor: \(134.149\)
Root analytic conductor: \(11.5822\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :21/2),\ 0.469 - 0.882i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.142648895\)
\(L(\frac12)\) \(\approx\) \(1.142648895\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.02e5 - 3.56e3i)T \)
good5 \( 1 + 3.58e7iT - 4.76e14T^{2} \)
7 \( 1 - 2.03e8iT - 5.58e17T^{2} \)
11 \( 1 - 1.20e11T + 7.40e21T^{2} \)
13 \( 1 - 1.77e11T + 2.47e23T^{2} \)
17 \( 1 - 1.07e13iT - 6.90e25T^{2} \)
19 \( 1 + 7.48e12iT - 7.14e26T^{2} \)
23 \( 1 - 2.50e14T + 3.94e28T^{2} \)
29 \( 1 + 1.36e15iT - 5.13e30T^{2} \)
31 \( 1 - 7.10e15iT - 2.08e31T^{2} \)
37 \( 1 + 5.30e16T + 8.55e32T^{2} \)
41 \( 1 - 1.12e17iT - 7.38e33T^{2} \)
43 \( 1 + 1.25e17iT - 2.00e34T^{2} \)
47 \( 1 - 1.01e17T + 1.30e35T^{2} \)
53 \( 1 - 6.36e17iT - 1.62e36T^{2} \)
59 \( 1 - 1.08e18T + 1.54e37T^{2} \)
61 \( 1 + 7.17e18T + 3.10e37T^{2} \)
67 \( 1 - 8.30e17iT - 2.22e38T^{2} \)
71 \( 1 + 3.40e19T + 7.52e38T^{2} \)
73 \( 1 + 6.94e18T + 1.34e39T^{2} \)
79 \( 1 - 5.54e19iT - 7.08e39T^{2} \)
83 \( 1 + 1.08e20T + 1.99e40T^{2} \)
89 \( 1 - 3.51e20iT - 8.65e40T^{2} \)
97 \( 1 + 7.50e20T + 5.27e41T^{2} \)
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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

−11.94777468442549303247726000551, −10.66642532383134455722638006412, −9.246887169981685017420034421768, −8.536928750610467631659469651948, −6.81881657644681647340263368381, −5.72526152032778145445210190049, −4.77476188920437588753608673355, −3.83523230930233363963705237714, −1.48071860075795935259342655480, −1.06729799794621397416862894727, 0.30520075619971287093727456850, 1.57477766738118748588328848627, 3.08109625325088449139456045548, 4.17167815729081795509616536870, 5.69700924116649476771400755474, 6.81310209397173744212466671448, 7.21572866784099297872588331455, 9.296620045424858882394401216018, 10.46346673722508449545810664208, 11.23872282881365861282304687499

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