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Properties

Label 2-48-16.3-c8-0-12
Degree $2$
Conductor $48$
Sign $0.392 + 0.919i$
Analytic cond. $19.5541$
Root an. cond. $4.42201$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.3 − 8.85i)2-s + (33.0 − 33.0i)3-s + (99.2 + 235. i)4-s + (−642. + 642. i)5-s + (−733. + 147. i)6-s − 3.32e3·7-s + (765. − 4.02e3i)8-s − 2.18e3i·9-s + (1.42e4 − 2.87e3i)10-s + (2.81e3 + 2.81e3i)11-s + (1.10e4 + 4.52e3i)12-s + (−2.76e3 − 2.76e3i)13-s + (4.42e4 + 2.94e4i)14-s + 4.24e4i·15-s + (−4.58e4 + 4.68e4i)16-s + 3.01e4·17-s + ⋯
L(s)  = 1  + (−0.832 − 0.553i)2-s + (0.408 − 0.408i)3-s + (0.387 + 0.921i)4-s + (−1.02 + 1.02i)5-s + (−0.565 + 0.114i)6-s − 1.38·7-s + (0.186 − 0.982i)8-s − 0.333i·9-s + (1.42 − 0.287i)10-s + (0.192 + 0.192i)11-s + (0.534 + 0.217i)12-s + (−0.0968 − 0.0968i)13-s + (1.15 + 0.765i)14-s + 0.839i·15-s + (−0.699 + 0.714i)16-s + 0.361·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.392 + 0.919i$
Analytic conductor: \(19.5541\)
Root analytic conductor: \(4.42201\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :4),\ 0.392 + 0.919i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.678530 - 0.447989i\)
\(L(\frac12)\) \(\approx\) \(0.678530 - 0.447989i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (13.3 + 8.85i)T \)
3 \( 1 + (-33.0 + 33.0i)T \)
good5 \( 1 + (642. - 642. i)T - 3.90e5iT^{2} \)
7 \( 1 + 3.32e3T + 5.76e6T^{2} \)
11 \( 1 + (-2.81e3 - 2.81e3i)T + 2.14e8iT^{2} \)
13 \( 1 + (2.76e3 + 2.76e3i)T + 8.15e8iT^{2} \)
17 \( 1 - 3.01e4T + 6.97e9T^{2} \)
19 \( 1 + (-1.06e5 + 1.06e5i)T - 1.69e10iT^{2} \)
23 \( 1 - 4.43e5T + 7.83e10T^{2} \)
29 \( 1 + (-3.14e5 - 3.14e5i)T + 5.00e11iT^{2} \)
31 \( 1 + 1.41e6iT - 8.52e11T^{2} \)
37 \( 1 + (1.67e6 - 1.67e6i)T - 3.51e12iT^{2} \)
41 \( 1 + 4.98e6iT - 7.98e12T^{2} \)
43 \( 1 + (2.03e6 + 2.03e6i)T + 1.16e13iT^{2} \)
47 \( 1 - 6.72e6iT - 2.38e13T^{2} \)
53 \( 1 + (-2.92e6 + 2.92e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (-1.69e7 - 1.69e7i)T + 1.46e14iT^{2} \)
61 \( 1 + (-2.92e6 - 2.92e6i)T + 1.91e14iT^{2} \)
67 \( 1 + (-1.68e7 + 1.68e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 8.31e5T + 6.45e14T^{2} \)
73 \( 1 + 9.05e6iT - 8.06e14T^{2} \)
79 \( 1 + 2.11e7iT - 1.51e15T^{2} \)
83 \( 1 + (1.73e7 - 1.73e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 9.27e6iT - 3.93e15T^{2} \)
97 \( 1 - 1.18e8T + 7.83e15T^{2} \)
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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

−13.43815879245029027953861058462, −12.33397262641382955493770116351, −11.31057212536667794149229562239, −10.08028404602550770795847565600, −8.931559598627575001315636253472, −7.40191378276883237012238841121, −6.80478064628678846853064535720, −3.55547149217561812393681100501, −2.78543716513681340928640639594, −0.53429149926672944112284978647, 0.878089760678205747062653239744, 3.38587121075191310402131417207, 5.14371134975310588908564126737, 6.87945746329052596731103150349, 8.217977830731708323680977293394, 9.141438266666508742036970826510, 10.12598657729173915678432747522, 11.71311489269526398691351943228, 12.97290589212082272815255886027, 14.49082158193760244071656942841

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