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Properties

Label 2-48-12.11-c21-0-1
Degree $2$
Conductor $48$
Sign $-0.296 + 0.954i$
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  + (−3.03e4 + 9.76e4i)3-s − 3.55e7i·5-s + 1.36e9i·7-s + (−8.61e9 − 5.93e9i)9-s − 8.13e10·11-s − 1.01e11·13-s + (3.47e12 + 1.07e12i)15-s + 1.39e13i·17-s + 9.45e12i·19-s + (−1.33e14 − 4.15e13i)21-s + 1.97e14·23-s − 7.87e14·25-s + (8.40e14 − 6.61e14i)27-s + 3.11e15i·29-s − 3.21e15i·31-s + ⋯
L(s)  = 1  + (−0.296 + 0.954i)3-s − 1.62i·5-s + 1.83i·7-s + (−0.823 − 0.566i)9-s − 0.945·11-s − 0.204·13-s + (1.55 + 0.483i)15-s + 1.67i·17-s + 0.353i·19-s + (−1.74 − 0.543i)21-s + 0.992·23-s − 1.65·25-s + (0.785 − 0.618i)27-s + 1.37i·29-s − 0.703i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.296 + 0.954i$
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: \((2,\ 48,\ (\ :21/2),\ -0.296 + 0.954i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.04887837066\)
\(L(\frac12)\) \(\approx\) \(0.04887837066\)
\(L(\frac{23}{2})\) 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 \( 1 + (3.03e4 - 9.76e4i)T \)
good5 \( 1 + 3.55e7iT - 4.76e14T^{2} \)
7 \( 1 - 1.36e9iT - 5.58e17T^{2} \)
11 \( 1 + 8.13e10T + 7.40e21T^{2} \)
13 \( 1 + 1.01e11T + 2.47e23T^{2} \)
17 \( 1 - 1.39e13iT - 6.90e25T^{2} \)
19 \( 1 - 9.45e12iT - 7.14e26T^{2} \)
23 \( 1 - 1.97e14T + 3.94e28T^{2} \)
29 \( 1 - 3.11e15iT - 5.13e30T^{2} \)
31 \( 1 + 3.21e15iT - 2.08e31T^{2} \)
37 \( 1 - 2.51e16T + 8.55e32T^{2} \)
41 \( 1 + 1.94e16iT - 7.38e33T^{2} \)
43 \( 1 - 2.27e16iT - 2.00e34T^{2} \)
47 \( 1 + 2.64e17T + 1.30e35T^{2} \)
53 \( 1 - 1.40e18iT - 1.62e36T^{2} \)
59 \( 1 - 1.50e17T + 1.54e37T^{2} \)
61 \( 1 + 1.64e17T + 3.10e37T^{2} \)
67 \( 1 - 1.03e19iT - 2.22e38T^{2} \)
71 \( 1 - 1.34e19T + 7.52e38T^{2} \)
73 \( 1 + 4.29e19T + 1.34e39T^{2} \)
79 \( 1 - 1.61e20iT - 7.08e39T^{2} \)
83 \( 1 + 1.59e20T + 1.99e40T^{2} \)
89 \( 1 + 4.43e20iT - 8.65e40T^{2} \)
97 \( 1 + 1.06e21T + 5.27e41T^{2} \)
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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

−12.43075197093427116399384163259, −11.19533410850289425159127678407, −9.799023032433908629898446770033, −8.812705277336515388114287201604, −8.324979458995409754322115964925, −5.83763876823804795892095510689, −5.34855667306327320391460646283, −4.38737211642353028003408914209, −2.86355703147060161999050210640, −1.51545503360940967240007088179, 0.01261921902352781797849856330, 0.841551588159459424720392939476, 2.42950257610684375285924951027, 3.23719734865617936722623262707, 4.87849088537678686318838309564, 6.50189653035037422636228108760, 7.22029687107939633932766247289, 7.73257953092691015366334683223, 9.937494674877697366795329319681, 10.85801349772372137463608247110

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