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Properties

Label 2-48-16.5-c5-0-15
Degree $2$
Conductor $48$
Sign $-0.964 - 0.263i$
Analytic cond. $7.69842$
Root an. cond. $2.77460$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.79 − 2.99i)2-s + (6.36 − 6.36i)3-s + (14.0 + 28.7i)4-s + (−26.7 − 26.7i)5-s + (−49.5 + 11.4i)6-s − 39.6i·7-s + (18.4 − 180. i)8-s − 81i·9-s + (48.3 + 208. i)10-s + (−195. − 195. i)11-s + (272. + 93.2i)12-s + (−440. + 440. i)13-s + (−118. + 190. i)14-s − 340.·15-s + (−627. + 809. i)16-s − 1.73e3·17-s + ⋯
L(s)  = 1  + (−0.848 − 0.529i)2-s + (0.408 − 0.408i)3-s + (0.439 + 0.898i)4-s + (−0.478 − 0.478i)5-s + (−0.562 + 0.130i)6-s − 0.306i·7-s + (0.101 − 0.994i)8-s − 0.333i·9-s + (0.152 + 0.659i)10-s + (−0.486 − 0.486i)11-s + (0.546 + 0.187i)12-s + (−0.723 + 0.723i)13-s + (−0.161 + 0.259i)14-s − 0.390·15-s + (−0.612 + 0.790i)16-s − 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.964 - 0.263i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ -0.964 - 0.263i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0568598 + 0.423313i\)
\(L(\frac12)\) \(\approx\) \(0.0568598 + 0.423313i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.79 + 2.99i)T \)
3 \( 1 + (-6.36 + 6.36i)T \)
good5 \( 1 + (26.7 + 26.7i)T + 3.12e3iT^{2} \)
7 \( 1 + 39.6iT - 1.68e4T^{2} \)
11 \( 1 + (195. + 195. i)T + 1.61e5iT^{2} \)
13 \( 1 + (440. - 440. i)T - 3.71e5iT^{2} \)
17 \( 1 + 1.73e3T + 1.41e6T^{2} \)
19 \( 1 + (1.08e3 - 1.08e3i)T - 2.47e6iT^{2} \)
23 \( 1 - 86.6iT - 6.43e6T^{2} \)
29 \( 1 + (-3.15e3 + 3.15e3i)T - 2.05e7iT^{2} \)
31 \( 1 + 2.54e3T + 2.86e7T^{2} \)
37 \( 1 + (-3.69e3 - 3.69e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.98e4iT - 1.15e8T^{2} \)
43 \( 1 + (7.11e3 + 7.11e3i)T + 1.47e8iT^{2} \)
47 \( 1 + 1.35e4T + 2.29e8T^{2} \)
53 \( 1 + (-9.12e3 - 9.12e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (-2.58e3 - 2.58e3i)T + 7.14e8iT^{2} \)
61 \( 1 + (-2.12e4 + 2.12e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (1.02e4 - 1.02e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 4.23e4iT - 1.80e9T^{2} \)
73 \( 1 + 5.49e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.07e5T + 3.07e9T^{2} \)
83 \( 1 + (7.22e4 - 7.22e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 8.25e4iT - 5.58e9T^{2} \)
97 \( 1 - 7.26e4T + 8.58e9T^{2} \)
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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

−13.77025242982837648996681443416, −12.65350672895562836045484426861, −11.66440654238039246828486453345, −10.38446193981432072020085951388, −8.956056800087669430205962947005, −8.073635572540537196588038639595, −6.79109268021908699472396975783, −4.13570044881399916233904652078, −2.22383170300524037450111869038, −0.25900233689792890514754434647, 2.53542973837772055559110518063, 4.90613944428494472470681745746, 6.77370632055986078867185081559, 7.957170334567252207556901510433, 9.132124616070544513071757446179, 10.35007042892364129032288951361, 11.32441423365628547735545128681, 13.10328467387819407610079122342, 14.83551084862049715465376185185, 15.17779717651477824679305155620

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