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Properties

Label 2-48-16.3-c6-0-2
Degree $2$
Conductor $48$
Sign $-0.874 - 0.485i$
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  + (−7.47 + 2.84i)2-s + (−11.0 + 11.0i)3-s + (47.7 − 42.5i)4-s + (−55.7 + 55.7i)5-s + (51.0 − 113. i)6-s + 496.·7-s + (−235. + 454. i)8-s − 242. i·9-s + (258. − 575. i)10-s + (694. + 694. i)11-s + (−57.0 + 996. i)12-s + (214. + 214. i)13-s + (−3.71e3 + 1.41e3i)14-s − 1.22e3i·15-s + (467. − 4.06e3i)16-s − 4.13e3·17-s + ⋯
L(s)  = 1  + (−0.934 + 0.356i)2-s + (−0.408 + 0.408i)3-s + (0.746 − 0.665i)4-s + (−0.446 + 0.446i)5-s + (0.236 − 0.526i)6-s + 1.44·7-s + (−0.460 + 0.887i)8-s − 0.333i·9-s + (0.258 − 0.575i)10-s + (0.521 + 0.521i)11-s + (−0.0330 + 0.576i)12-s + (0.0976 + 0.0976i)13-s + (−1.35 + 0.515i)14-s − 0.364i·15-s + (0.114 − 0.993i)16-s − 0.840·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.874 - 0.485i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ -0.874 - 0.485i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.173765 + 0.670449i\)
\(L(\frac12)\) \(\approx\) \(0.173765 + 0.670449i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.47 - 2.84i)T \)
3 \( 1 + (11.0 - 11.0i)T \)
good5 \( 1 + (55.7 - 55.7i)T - 1.56e4iT^{2} \)
7 \( 1 - 496.T + 1.17e5T^{2} \)
11 \( 1 + (-694. - 694. i)T + 1.77e6iT^{2} \)
13 \( 1 + (-214. - 214. i)T + 4.82e6iT^{2} \)
17 \( 1 + 4.13e3T + 2.41e7T^{2} \)
19 \( 1 + (7.22e3 - 7.22e3i)T - 4.70e7iT^{2} \)
23 \( 1 + 2.02e4T + 1.48e8T^{2} \)
29 \( 1 + (-2.57e4 - 2.57e4i)T + 5.94e8iT^{2} \)
31 \( 1 + 3.55e4iT - 8.87e8T^{2} \)
37 \( 1 + (1.20e4 - 1.20e4i)T - 2.56e9iT^{2} \)
41 \( 1 - 5.59e3iT - 4.75e9T^{2} \)
43 \( 1 + (1.22e4 + 1.22e4i)T + 6.32e9iT^{2} \)
47 \( 1 - 1.26e5iT - 1.07e10T^{2} \)
53 \( 1 + (6.91e4 - 6.91e4i)T - 2.21e10iT^{2} \)
59 \( 1 + (1.01e5 + 1.01e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (-1.23e5 - 1.23e5i)T + 5.15e10iT^{2} \)
67 \( 1 + (3.94e5 - 3.94e5i)T - 9.04e10iT^{2} \)
71 \( 1 + 2.64e5T + 1.28e11T^{2} \)
73 \( 1 - 2.30e5iT - 1.51e11T^{2} \)
79 \( 1 + 2.78e5iT - 2.43e11T^{2} \)
83 \( 1 + (-6.05e5 + 6.05e5i)T - 3.26e11iT^{2} \)
89 \( 1 + 5.26e5iT - 4.96e11T^{2} \)
97 \( 1 - 8.40e5T + 8.32e11T^{2} \)
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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

−14.98635261236895033483193269554, −14.36858049690201214283035800223, −11.98851568091488356060732966837, −11.15333490926374323447044245339, −10.21120682862857465917145785498, −8.687794794231992421539988500956, −7.59689254052520695862573805523, −6.16077777369082650489396080105, −4.43600170918702775890777619403, −1.74110081407488268741255564059, 0.44678802828524593415273565965, 1.96003538434567129571929453357, 4.43452086326435736444555911614, 6.50094465577933528849106757583, 8.045379265374842130408816434952, 8.692324625747649066546478430573, 10.57482603079980044401583798575, 11.51030554297933631135617325198, 12.24393100543951920242859227308, 13.79601480274251734327519410986

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