Properties

Label 2-48-16.11-c6-0-17
Degree $2$
Conductor $48$
Sign $-0.350 + 0.936i$
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  + (−4.50 − 6.61i)2-s + (11.0 + 11.0i)3-s + (−23.4 + 59.5i)4-s + (−9.26 − 9.26i)5-s + (23.2 − 122. i)6-s − 320.·7-s + (499. − 112. i)8-s + 242. i·9-s + (−19.5 + 103. i)10-s + (1.49e3 − 1.49e3i)11-s + (−914. + 397. i)12-s + (2.54e3 − 2.54e3i)13-s + (1.44e3 + 2.11e3i)14-s − 204. i·15-s + (−2.99e3 − 2.79e3i)16-s − 7.23e3·17-s + ⋯
L(s)  = 1  + (−0.562 − 0.826i)2-s + (0.408 + 0.408i)3-s + (−0.366 + 0.930i)4-s + (−0.0741 − 0.0741i)5-s + (0.107 − 0.567i)6-s − 0.934·7-s + (0.975 − 0.220i)8-s + 0.333i·9-s + (−0.0195 + 0.103i)10-s + (1.12 − 1.12i)11-s + (−0.529 + 0.230i)12-s + (1.15 − 1.15i)13-s + (0.525 + 0.772i)14-s − 0.0605i·15-s + (−0.731 − 0.682i)16-s − 1.47·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.350 + 0.936i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ -0.350 + 0.936i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.635260 - 0.915860i\)
\(L(\frac12)\) \(\approx\) \(0.635260 - 0.915860i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.50 + 6.61i)T \)
3 \( 1 + (-11.0 - 11.0i)T \)
good5 \( 1 + (9.26 + 9.26i)T + 1.56e4iT^{2} \)
7 \( 1 + 320.T + 1.17e5T^{2} \)
11 \( 1 + (-1.49e3 + 1.49e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (-2.54e3 + 2.54e3i)T - 4.82e6iT^{2} \)
17 \( 1 + 7.23e3T + 2.41e7T^{2} \)
19 \( 1 + (4.86e3 + 4.86e3i)T + 4.70e7iT^{2} \)
23 \( 1 - 1.22e4T + 1.48e8T^{2} \)
29 \( 1 + (-9.52e3 + 9.52e3i)T - 5.94e8iT^{2} \)
31 \( 1 + 1.35e4iT - 8.87e8T^{2} \)
37 \( 1 + (964. + 964. i)T + 2.56e9iT^{2} \)
41 \( 1 - 9.46e4iT - 4.75e9T^{2} \)
43 \( 1 + (-6.14e4 + 6.14e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 1.25e5iT - 1.07e10T^{2} \)
53 \( 1 + (-9.27e4 - 9.27e4i)T + 2.21e10iT^{2} \)
59 \( 1 + (7.89e4 - 7.89e4i)T - 4.21e10iT^{2} \)
61 \( 1 + (-1.40e4 + 1.40e4i)T - 5.15e10iT^{2} \)
67 \( 1 + (1.69e5 + 1.69e5i)T + 9.04e10iT^{2} \)
71 \( 1 - 2.39e5T + 1.28e11T^{2} \)
73 \( 1 - 6.68e5iT - 1.51e11T^{2} \)
79 \( 1 + 5.72e5iT - 2.43e11T^{2} \)
83 \( 1 + (3.60e5 + 3.60e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 1.01e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.29e6T + 8.32e11T^{2} \)
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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

−13.63202054701473441677907134495, −12.97688947855031912281982934073, −11.38393095461266609925280501712, −10.50673759872512601423563694528, −9.092862292419090076456462218652, −8.471863511405099432112325547670, −6.47021615119199657095307693450, −4.06075174433573952450096491377, −2.88064342159563408062237284056, −0.61443094347938687332655214388, 1.59775341222094066136884341809, 4.15370546942672452512336690908, 6.43398765749147827663945744536, 6.99143606372750268598805660783, 8.799293224283402680712141444447, 9.414770931688585603739472740918, 11.03485165835466561375944385729, 12.71693819010871780580629801465, 13.82578106152467092674876644662, 14.87812022835028875063571587999

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