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Properties

Label 2-48-12.11-c21-0-8
Degree $2$
Conductor $48$
Sign $-0.923 + 0.384i$
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  + (−9.44e4 + 3.93e4i)3-s + 2.52e7i·5-s + 1.86e8i·7-s + (7.36e9 − 7.42e9i)9-s − 7.28e10·11-s + 4.38e11·13-s + (−9.92e11 − 2.38e12i)15-s + 5.98e12i·17-s − 4.42e12i·19-s + (−7.32e12 − 1.75e13i)21-s − 4.50e13·23-s − 1.60e14·25-s + (−4.03e14 + 9.90e14i)27-s + 1.46e15i·29-s + 7.97e15i·31-s + ⋯
L(s)  = 1  + (−0.923 + 0.384i)3-s + 1.15i·5-s + 0.249i·7-s + (0.704 − 0.709i)9-s − 0.847·11-s + 0.881·13-s + (−0.444 − 1.06i)15-s + 0.719i·17-s − 0.165i·19-s + (−0.0958 − 0.230i)21-s − 0.226·23-s − 0.337·25-s + (−0.377 + 0.925i)27-s + 0.646i·29-s + 1.74i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.923 + 0.384i$
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: \((2,\ 48,\ (\ :21/2),\ -0.923 + 0.384i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.9408142088\)
\(L(\frac12)\) \(\approx\) \(0.9408142088\)
\(L(\frac{23}{2})\) 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 \( 1 + (9.44e4 - 3.93e4i)T \)
good5 \( 1 - 2.52e7iT - 4.76e14T^{2} \)
7 \( 1 - 1.86e8iT - 5.58e17T^{2} \)
11 \( 1 + 7.28e10T + 7.40e21T^{2} \)
13 \( 1 - 4.38e11T + 2.47e23T^{2} \)
17 \( 1 - 5.98e12iT - 6.90e25T^{2} \)
19 \( 1 + 4.42e12iT - 7.14e26T^{2} \)
23 \( 1 + 4.50e13T + 3.94e28T^{2} \)
29 \( 1 - 1.46e15iT - 5.13e30T^{2} \)
31 \( 1 - 7.97e15iT - 2.08e31T^{2} \)
37 \( 1 + 8.35e15T + 8.55e32T^{2} \)
41 \( 1 - 2.72e16iT - 7.38e33T^{2} \)
43 \( 1 - 1.09e16iT - 2.00e34T^{2} \)
47 \( 1 - 4.14e17T + 1.30e35T^{2} \)
53 \( 1 - 1.78e18iT - 1.62e36T^{2} \)
59 \( 1 + 4.28e17T + 1.54e37T^{2} \)
61 \( 1 - 9.27e18T + 3.10e37T^{2} \)
67 \( 1 - 2.56e19iT - 2.22e38T^{2} \)
71 \( 1 + 2.34e19T + 7.52e38T^{2} \)
73 \( 1 + 2.85e19T + 1.34e39T^{2} \)
79 \( 1 + 8.48e19iT - 7.08e39T^{2} \)
83 \( 1 - 6.81e19T + 1.99e40T^{2} \)
89 \( 1 - 8.17e19iT - 8.65e40T^{2} \)
97 \( 1 - 8.73e19T + 5.27e41T^{2} \)
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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

−11.89706638358042022488299535591, −10.67472479282208561884911123669, −10.44001067481997858981942857680, −8.788998446740630988998593031334, −7.24152367756629082212118293195, −6.27374978611485839270093638410, −5.30873921957262614165962958102, −3.88884618569952613100710740537, −2.77933448513920839951228750860, −1.24098748078689393685106720640, 0.27959460553520925731728736562, 0.909766877781805063751716200134, 2.16397109507900781290019017780, 4.07210821064700911603026092197, 5.12603051425272500356325309110, 5.97820894514241190293416663534, 7.39635720954430083777033926750, 8.435471049183637003482387185615, 9.814360348252139938509761802185, 11.00265889829979756281510473477

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