Properties

Label 2-48-4.3-c8-0-4
Degree $2$
Conductor $48$
Sign $0.866 + 0.5i$
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  + 46.7i·3-s − 1.08e3·5-s + 426. i·7-s − 2.18e3·9-s − 1.42e4i·11-s + 3.42e4·13-s − 5.07e4i·15-s + 2.00e4·17-s + 1.96e5i·19-s − 1.99e4·21-s − 3.47e5i·23-s + 7.85e5·25-s − 1.02e5i·27-s + 1.00e6·29-s − 1.63e6i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.73·5-s + 0.177i·7-s − 0.333·9-s − 0.972i·11-s + 1.19·13-s − 1.00i·15-s + 0.240·17-s + 1.50i·19-s − 0.102·21-s − 1.24i·23-s + 2.01·25-s − 0.192i·27-s + 1.41·29-s − 1.77i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(19.5541\)
Root analytic conductor: \(4.42201\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :4),\ 0.866 + 0.5i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.08609 - 0.291017i\)
\(L(\frac12)\) \(\approx\) \(1.08609 - 0.291017i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 46.7iT \)
good5 \( 1 + 1.08e3T + 3.90e5T^{2} \)
7 \( 1 - 426. iT - 5.76e6T^{2} \)
11 \( 1 + 1.42e4iT - 2.14e8T^{2} \)
13 \( 1 - 3.42e4T + 8.15e8T^{2} \)
17 \( 1 - 2.00e4T + 6.97e9T^{2} \)
19 \( 1 - 1.96e5iT - 1.69e10T^{2} \)
23 \( 1 + 3.47e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.00e6T + 5.00e11T^{2} \)
31 \( 1 + 1.63e6iT - 8.52e11T^{2} \)
37 \( 1 + 7.91e5T + 3.51e12T^{2} \)
41 \( 1 - 1.36e6T + 7.98e12T^{2} \)
43 \( 1 + 1.50e6iT - 1.16e13T^{2} \)
47 \( 1 + 1.49e6iT - 2.38e13T^{2} \)
53 \( 1 + 8.94e6T + 6.22e13T^{2} \)
59 \( 1 + 8.50e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.78e7T + 1.91e14T^{2} \)
67 \( 1 - 3.49e7iT - 4.06e14T^{2} \)
71 \( 1 + 3.84e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.11e7T + 8.06e14T^{2} \)
79 \( 1 + 3.67e7iT - 1.51e15T^{2} \)
83 \( 1 - 2.94e7iT - 2.25e15T^{2} \)
89 \( 1 + 3.70e7T + 3.93e15T^{2} \)
97 \( 1 - 1.26e8T + 7.83e15T^{2} \)
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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

−14.04008404465898606345051473219, −12.38800106421394934501369701395, −11.43159702826250362470738162734, −10.51340162150770056522842181820, −8.655488273495206604475835029838, −7.955576637442554500544988710394, −6.09316296515965367551517464232, −4.26293231391293710499151709306, −3.32642054303944437633794558853, −0.54801821088576392674407038783, 1.02825423984603466199324502254, 3.26510463481869745562304685298, 4.66787507895472621210964477443, 6.79758137194928541844273173034, 7.71590481882873833704085154563, 8.855077400715678482763704031034, 10.80873749054483916959568649379, 11.76645393164492708947681141207, 12.67235611460835083556241419751, 13.97171747309483504971428597876

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