# Properties

 Label 2-48-3.2-c12-0-5 Degree $2$ Conductor $48$ Sign $-0.612 - 0.790i$ Analytic cond. $43.8717$ Root an. cond. $6.62357$ Motivic weight $12$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (446. + 576. i)3-s − 1.16e4i·5-s + 2.42e4·7-s + (−1.32e5 + 5.14e5i)9-s − 1.35e6i·11-s − 5.79e6·13-s + (6.70e6 − 5.19e6i)15-s + 4.63e7i·17-s + 3.36e7·19-s + (1.08e7 + 1.39e7i)21-s + 1.86e8i·23-s + 1.08e8·25-s + (−3.55e8 + 1.53e8i)27-s − 2.36e8i·29-s + 5.29e7·31-s + ⋯
 L(s)  = 1 + (0.612 + 0.790i)3-s − 0.744i·5-s + 0.205·7-s + (−0.248 + 0.968i)9-s − 0.764i·11-s − 1.19·13-s + (0.588 − 0.456i)15-s + 1.92i·17-s + 0.714·19-s + (0.126 + 0.162i)21-s + 1.25i·23-s + 0.445·25-s + (−0.917 + 0.396i)27-s − 0.397i·29-s + 0.0596·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(13-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$48$$    =    $$2^{4} \cdot 3$$ Sign: $-0.612 - 0.790i$ Analytic conductor: $$43.8717$$ Root analytic conductor: $$6.62357$$ Motivic weight: $$12$$ Rational: no Arithmetic: yes Character: $\chi_{48} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 48,\ (\ :6),\ -0.612 - 0.790i)$$

## Particular Values

 $$L(\frac{13}{2})$$ $$\approx$$ $$1.612776751$$ $$L(\frac12)$$ $$\approx$$ $$1.612776751$$ $$L(7)$$ 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 + (-446. - 576. i)T$$
good5 $$1 + 1.16e4iT - 2.44e8T^{2}$$
7 $$1 - 2.42e4T + 1.38e10T^{2}$$
11 $$1 + 1.35e6iT - 3.13e12T^{2}$$
13 $$1 + 5.79e6T + 2.32e13T^{2}$$
17 $$1 - 4.63e7iT - 5.82e14T^{2}$$
19 $$1 - 3.36e7T + 2.21e15T^{2}$$
23 $$1 - 1.86e8iT - 2.19e16T^{2}$$
29 $$1 + 2.36e8iT - 3.53e17T^{2}$$
31 $$1 - 5.29e7T + 7.87e17T^{2}$$
37 $$1 + 4.83e9T + 6.58e18T^{2}$$
41 $$1 - 6.23e9iT - 2.25e19T^{2}$$
43 $$1 - 6.78e9T + 3.99e19T^{2}$$
47 $$1 - 1.19e10iT - 1.16e20T^{2}$$
53 $$1 - 7.56e9iT - 4.91e20T^{2}$$
59 $$1 - 5.06e10iT - 1.77e21T^{2}$$
61 $$1 + 5.11e10T + 2.65e21T^{2}$$
67 $$1 + 9.11e9T + 8.18e21T^{2}$$
71 $$1 + 1.39e11iT - 1.64e22T^{2}$$
73 $$1 + 6.80e9T + 2.29e22T^{2}$$
79 $$1 + 3.21e11T + 5.90e22T^{2}$$
83 $$1 - 5.38e11iT - 1.06e23T^{2}$$
89 $$1 - 5.38e10iT - 2.46e23T^{2}$$
97 $$1 - 7.08e10T + 6.93e23T^{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

−13.58327058775724209958350749114, −12.37685600300538789554826835372, −10.95715163799240930907632092671, −9.779582374289739856884442042617, −8.726750491806532863798388282522, −7.73888531989805003660794583344, −5.64378025292469873144710820413, −4.49248424043352922792303746202, −3.17840489616138815479489636266, −1.52704178392593821537966344923, 0.39375323377761732490860580526, 2.12211715443788773456193289942, 3.06282025190465444793469390177, 4.97376784916328939470148139548, 6.92510922435535049576430673573, 7.38285847873404560347430529967, 8.983791854760127952955278220814, 10.15151941847501005813197630347, 11.73140786485483580124006336331, 12.58430478030809969028517986751