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Properties

Label 2-48-16.11-c6-0-1
Degree $2$
Conductor $48$
Sign $-0.297 - 0.954i$
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  + (0.181 − 7.99i)2-s + (11.0 + 11.0i)3-s + (−63.9 − 2.89i)4-s + (−124. − 124. i)5-s + (90.1 − 86.1i)6-s + 200.·7-s + (−34.7 + 510. i)8-s + 242. i·9-s + (−1.01e3 + 973. i)10-s + (−432. + 432. i)11-s + (−672. − 736. i)12-s + (−2.45e3 + 2.45e3i)13-s + (36.2 − 1.60e3i)14-s − 2.74e3i·15-s + (4.07e3 + 370. i)16-s − 8.60e3·17-s + ⋯
L(s)  = 1  + (0.0226 − 0.999i)2-s + (0.408 + 0.408i)3-s + (−0.998 − 0.0452i)4-s + (−0.996 − 0.996i)5-s + (0.417 − 0.398i)6-s + 0.583·7-s + (−0.0678 + 0.997i)8-s + 0.333i·9-s + (−1.01 + 0.973i)10-s + (−0.325 + 0.325i)11-s + (−0.389 − 0.426i)12-s + (−1.11 + 1.11i)13-s + (0.0132 − 0.583i)14-s − 0.813i·15-s + (0.995 + 0.0904i)16-s − 1.75·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.297 - 0.954i$
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: \((2,\ 48,\ (\ :3),\ -0.297 - 0.954i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0631538 + 0.0858322i\)
\(L(\frac12)\) \(\approx\) \(0.0631538 + 0.0858322i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.181 + 7.99i)T \)
3 \( 1 + (-11.0 - 11.0i)T \)
good5 \( 1 + (124. + 124. i)T + 1.56e4iT^{2} \)
7 \( 1 - 200.T + 1.17e5T^{2} \)
11 \( 1 + (432. - 432. i)T - 1.77e6iT^{2} \)
13 \( 1 + (2.45e3 - 2.45e3i)T - 4.82e6iT^{2} \)
17 \( 1 + 8.60e3T + 2.41e7T^{2} \)
19 \( 1 + (-6.17e3 - 6.17e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 1.02e4T + 1.48e8T^{2} \)
29 \( 1 + (-1.68e4 + 1.68e4i)T - 5.94e8iT^{2} \)
31 \( 1 + 2.87e4iT - 8.87e8T^{2} \)
37 \( 1 + (3.08e4 + 3.08e4i)T + 2.56e9iT^{2} \)
41 \( 1 + 3.19e4iT - 4.75e9T^{2} \)
43 \( 1 + (9.77e4 - 9.77e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 1.14e5iT - 1.07e10T^{2} \)
53 \( 1 + (-9.00e4 - 9.00e4i)T + 2.21e10iT^{2} \)
59 \( 1 + (-7.89e3 + 7.89e3i)T - 4.21e10iT^{2} \)
61 \( 1 + (-3.76e4 + 3.76e4i)T - 5.15e10iT^{2} \)
67 \( 1 + (1.92e5 + 1.92e5i)T + 9.04e10iT^{2} \)
71 \( 1 + 4.16e5T + 1.28e11T^{2} \)
73 \( 1 - 1.95e5iT - 1.51e11T^{2} \)
79 \( 1 - 3.83e5iT - 2.43e11T^{2} \)
83 \( 1 + (-1.26e5 - 1.26e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 4.98e5iT - 4.96e11T^{2} \)
97 \( 1 - 7.38e5T + 8.32e11T^{2} \)
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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

−14.59542558159192088218912754373, −13.44774702017198986619653170384, −12.12750416433899296510757615114, −11.49379790253950232428740897102, −9.932079902972245700577506259087, −8.827567997174856812681473430472, −7.81545353993006939143385118129, −4.83154784051720459646541554803, −4.12262358129864714145467022902, −2.03464664462865601451048534041, 0.04430297985006192911290386721, 3.08706769878198163826657862656, 4.87142772135635978425510681378, 6.80103613657262688148822520663, 7.62524755178792811214306286491, 8.628581830800099444727617156470, 10.40351312444803809657767566268, 11.82579461644286951815171775773, 13.25395449751869746319371424086, 14.34882136206750892332666036100

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