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Properties

Label 2-48-12.11-c21-0-33
Degree $2$
Conductor $48$
Sign $-0.868 + 0.495i$
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  + (554. − 1.02e5i)3-s − 3.49e7i·5-s + 7.47e8i·7-s + (−1.04e10 − 1.13e8i)9-s + 1.00e11·11-s + 2.55e11·13-s + (−3.57e12 − 1.93e10i)15-s − 6.80e12i·17-s + 4.43e13i·19-s + (7.64e13 + 4.14e11i)21-s + 2.67e14·23-s − 7.46e14·25-s + (−1.73e13 + 1.06e15i)27-s − 4.19e15i·29-s − 4.81e15i·31-s + ⋯
L(s)  = 1  + (0.00541 − 0.999i)3-s − 1.60i·5-s + 1.00i·7-s + (−0.999 − 0.0108i)9-s + 1.16·11-s + 0.513·13-s + (−1.60 − 0.00868i)15-s − 0.818i·17-s + 1.65i·19-s + (1.00 + 0.00542i)21-s + 1.34·23-s − 1.56·25-s + (−0.0162 + 0.999i)27-s − 1.85i·29-s − 1.05i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.868 + 0.495i$
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: \((2,\ 48,\ (\ :21/2),\ -0.868 + 0.495i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.285019893\)
\(L(\frac12)\) \(\approx\) \(2.285019893\)
\(L(\frac{23}{2})\) 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 \( 1 + (-554. + 1.02e5i)T \)
good5 \( 1 + 3.49e7iT - 4.76e14T^{2} \)
7 \( 1 - 7.47e8iT - 5.58e17T^{2} \)
11 \( 1 - 1.00e11T + 7.40e21T^{2} \)
13 \( 1 - 2.55e11T + 2.47e23T^{2} \)
17 \( 1 + 6.80e12iT - 6.90e25T^{2} \)
19 \( 1 - 4.43e13iT - 7.14e26T^{2} \)
23 \( 1 - 2.67e14T + 3.94e28T^{2} \)
29 \( 1 + 4.19e15iT - 5.13e30T^{2} \)
31 \( 1 + 4.81e15iT - 2.08e31T^{2} \)
37 \( 1 - 3.54e16T + 8.55e32T^{2} \)
41 \( 1 + 1.43e17iT - 7.38e33T^{2} \)
43 \( 1 + 1.16e17iT - 2.00e34T^{2} \)
47 \( 1 + 1.48e17T + 1.30e35T^{2} \)
53 \( 1 - 1.61e18iT - 1.62e36T^{2} \)
59 \( 1 - 1.11e18T + 1.54e37T^{2} \)
61 \( 1 - 1.49e18T + 3.10e37T^{2} \)
67 \( 1 + 1.07e19iT - 2.22e38T^{2} \)
71 \( 1 - 2.20e19T + 7.52e38T^{2} \)
73 \( 1 + 3.19e19T + 1.34e39T^{2} \)
79 \( 1 - 2.21e19iT - 7.08e39T^{2} \)
83 \( 1 + 2.26e19T + 1.99e40T^{2} \)
89 \( 1 - 2.74e20iT - 8.65e40T^{2} \)
97 \( 1 - 3.33e20T + 5.27e41T^{2} \)
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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

−11.60428069556469959019486098781, −9.371975281650764390201190721968, −8.744770128623848881426068765526, −7.75763332069864616515105876608, −6.17072770824767845776700771380, −5.42398718957767300671072908764, −3.98926290253835908860631854110, −2.30469159449720624500840619247, −1.28563090220354546950225333409, −0.53304939540624989637953001194, 1.11467667178940975607125336468, 2.92825478467541143411346148524, 3.57787716582040270219758141733, 4.70303834021454276687038962871, 6.41156941778495680908060854290, 7.06561283015552685728118463650, 8.746918578364959453866670473186, 9.905741397365650093663385929768, 11.00531524814112859604893438515, 11.19032809436993762757280832602

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