Properties

Label 2-48-4.3-c6-0-2
Degree $2$
Conductor $48$
Sign $0.866 - 0.5i$
Analytic cond. $11.0425$
Root an. cond. $3.32304$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.5i·3-s + 150·5-s − 325. i·7-s − 243·9-s + 1.47e3i·11-s + 3.39e3·13-s + 2.33e3i·15-s + 5.17e3·17-s + 6.81e3i·19-s + 5.07e3·21-s − 3.99e3i·23-s + 6.87e3·25-s − 3.78e3i·27-s + 3.21e4·29-s + 3.26e4i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.19·5-s − 0.949i·7-s − 0.333·9-s + 1.10i·11-s + 1.54·13-s + 0.692i·15-s + 1.05·17-s + 0.992i·19-s + 0.548·21-s − 0.327i·23-s + 0.440·25-s − 0.192i·27-s + 1.31·29-s + 1.09i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ 0.866 - 0.5i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.14124 + 0.573743i\)
\(L(\frac12)\) \(\approx\) \(2.14124 + 0.573743i\)
\(L(4)\) 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 \( 1 - 15.5iT \)
good5 \( 1 - 150T + 1.56e4T^{2} \)
7 \( 1 + 325. iT - 1.17e5T^{2} \)
11 \( 1 - 1.47e3iT - 1.77e6T^{2} \)
13 \( 1 - 3.39e3T + 4.82e6T^{2} \)
17 \( 1 - 5.17e3T + 2.41e7T^{2} \)
19 \( 1 - 6.81e3iT - 4.70e7T^{2} \)
23 \( 1 + 3.99e3iT - 1.48e8T^{2} \)
29 \( 1 - 3.21e4T + 5.94e8T^{2} \)
31 \( 1 - 3.26e4iT - 8.87e8T^{2} \)
37 \( 1 + 7.61e4T + 2.56e9T^{2} \)
41 \( 1 + 7.00e4T + 4.75e9T^{2} \)
43 \( 1 + 1.00e5iT - 6.32e9T^{2} \)
47 \( 1 + 1.51e5iT - 1.07e10T^{2} \)
53 \( 1 - 6.69e4T + 2.21e10T^{2} \)
59 \( 1 + 3.90e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.57e5T + 5.15e10T^{2} \)
67 \( 1 - 3.21e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.43e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.43e5T + 1.51e11T^{2} \)
79 \( 1 - 4.74e5iT - 2.43e11T^{2} \)
83 \( 1 + 1.03e6iT - 3.26e11T^{2} \)
89 \( 1 + 6.86e5T + 4.96e11T^{2} \)
97 \( 1 + 9.42e5T + 8.32e11T^{2} \)
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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

−14.25903235033281588114271412315, −13.64964898334371702673047700766, −12.23555806775937684344234477076, −10.41384330456026125450276741105, −10.07557653603962168161161801631, −8.531351005914113195887810404574, −6.75642421751498032990890837195, −5.34542572234888493888685663098, −3.69904105323943344519986996636, −1.51022245345459716902164341384, 1.28406475732437155487881840201, 2.92924760853123386070338710136, 5.59815303067403071130218220833, 6.31229452120245479164839927420, 8.303033359169925293591242609885, 9.284074084583963413651739528205, 10.82473273115840995688584584309, 12.05056755219228819680087407418, 13.43299135497336254025284132279, 13.89140620004396979767327969704

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