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Properties

Label 2-48-16.11-c6-0-13
Degree $2$
Conductor $48$
Sign $0.551 + 0.834i$
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  + (−0.382 − 7.99i)2-s + (−11.0 − 11.0i)3-s + (−63.7 + 6.11i)4-s + (122. + 122. i)5-s + (−83.8 + 92.2i)6-s + 392.·7-s + (73.2 + 506. i)8-s + 242. i·9-s + (930. − 1.02e3i)10-s + (469. − 469. i)11-s + (769. + 634. i)12-s + (284. − 284. i)13-s + (−150. − 3.13e3i)14-s − 2.69e3i·15-s + (4.02e3 − 779. i)16-s − 4.54e3·17-s + ⋯
L(s)  = 1  + (−0.0478 − 0.998i)2-s + (−0.408 − 0.408i)3-s + (−0.995 + 0.0955i)4-s + (0.978 + 0.978i)5-s + (−0.388 + 0.427i)6-s + 1.14·7-s + (0.143 + 0.989i)8-s + 0.333i·9-s + (0.930 − 1.02i)10-s + (0.352 − 0.352i)11-s + (0.445 + 0.367i)12-s + (0.129 − 0.129i)13-s + (−0.0547 − 1.14i)14-s − 0.799i·15-s + (0.981 − 0.190i)16-s − 0.924·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.551 + 0.834i$
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: \((2,\ 48,\ (\ :3),\ 0.551 + 0.834i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.58956 - 0.854723i\)
\(L(\frac12)\) \(\approx\) \(1.58956 - 0.854723i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 + 7.99i)T \)
3 \( 1 + (11.0 + 11.0i)T \)
good5 \( 1 + (-122. - 122. i)T + 1.56e4iT^{2} \)
7 \( 1 - 392.T + 1.17e5T^{2} \)
11 \( 1 + (-469. + 469. i)T - 1.77e6iT^{2} \)
13 \( 1 + (-284. + 284. i)T - 4.82e6iT^{2} \)
17 \( 1 + 4.54e3T + 2.41e7T^{2} \)
19 \( 1 + (-2.14e3 - 2.14e3i)T + 4.70e7iT^{2} \)
23 \( 1 - 2.34e4T + 1.48e8T^{2} \)
29 \( 1 + (-2.97e4 + 2.97e4i)T - 5.94e8iT^{2} \)
31 \( 1 + 6.95e3iT - 8.87e8T^{2} \)
37 \( 1 + (-4.28e4 - 4.28e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 5.10e4iT - 4.75e9T^{2} \)
43 \( 1 + (5.65e4 - 5.65e4i)T - 6.32e9iT^{2} \)
47 \( 1 - 6.21e4iT - 1.07e10T^{2} \)
53 \( 1 + (1.75e5 + 1.75e5i)T + 2.21e10iT^{2} \)
59 \( 1 + (-1.47e5 + 1.47e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (1.89e5 - 1.89e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (-5.92e4 - 5.92e4i)T + 9.04e10iT^{2} \)
71 \( 1 - 5.75e5T + 1.28e11T^{2} \)
73 \( 1 + 3.77e5iT - 1.51e11T^{2} \)
79 \( 1 - 6.05e5iT - 2.43e11T^{2} \)
83 \( 1 + (6.44e5 + 6.44e5i)T + 3.26e11iT^{2} \)
89 \( 1 - 6.51e5iT - 4.96e11T^{2} \)
97 \( 1 + 4.63e5T + 8.32e11T^{2} \)
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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

−13.95985894656981487160270131234, −13.10144655098125077490819879609, −11.48632441861765532402563587012, −10.97711477582980273054422747944, −9.723063093124671485217138848903, −8.203348770908321453677887827506, −6.41307934197201298538957376274, −4.86304457878420211240357898615, −2.69193776526610605055915252156, −1.29659062654183791929314563638, 1.18934534701715772956914349806, 4.63933239359735809293912099087, 5.31021977642538643768802369788, 6.85933596587431625399634634567, 8.635161996453428510441129345585, 9.349934180962724964506868372749, 10.87805564988738135871168492212, 12.54068801137217038369887408838, 13.65766688044907234329151816346, 14.69036956470208028530342958938

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