Properties

Label 2-48-3.2-c12-0-12
Degree $2$
Conductor $48$
Sign $0.818 - 0.574i$
Analytic cond. $43.8717$
Root an. cond. $6.62357$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−596. + 418. i)3-s + 2.39e4i·5-s − 3.21e4·7-s + (1.80e5 − 4.99e5i)9-s − 2.66e6i·11-s + 7.50e6·13-s + (−1.00e7 − 1.42e7i)15-s + 1.32e6i·17-s − 3.25e7·19-s + (1.91e7 − 1.34e7i)21-s − 8.48e7i·23-s − 3.27e8·25-s + (1.01e8 + 3.73e8i)27-s − 8.40e8i·29-s + 1.20e9·31-s + ⋯
L(s)  = 1  + (−0.818 + 0.574i)3-s + 1.53i·5-s − 0.273·7-s + (0.340 − 0.940i)9-s − 1.50i·11-s + 1.55·13-s + (−0.878 − 1.25i)15-s + 0.0550i·17-s − 0.692·19-s + (0.223 − 0.156i)21-s − 0.573i·23-s − 1.34·25-s + (0.261 + 0.965i)27-s − 1.41i·29-s + 1.35·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.818 - 0.574i$
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: \((2,\ 48,\ (\ :6),\ 0.818 - 0.574i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.455542112\)
\(L(\frac12)\) \(\approx\) \(1.455542112\)
\(L(7)\) 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 \( 1 + (596. - 418. i)T \)
good5 \( 1 - 2.39e4iT - 2.44e8T^{2} \)
7 \( 1 + 3.21e4T + 1.38e10T^{2} \)
11 \( 1 + 2.66e6iT - 3.13e12T^{2} \)
13 \( 1 - 7.50e6T + 2.32e13T^{2} \)
17 \( 1 - 1.32e6iT - 5.82e14T^{2} \)
19 \( 1 + 3.25e7T + 2.21e15T^{2} \)
23 \( 1 + 8.48e7iT - 2.19e16T^{2} \)
29 \( 1 + 8.40e8iT - 3.53e17T^{2} \)
31 \( 1 - 1.20e9T + 7.87e17T^{2} \)
37 \( 1 - 1.09e9T + 6.58e18T^{2} \)
41 \( 1 - 5.50e9iT - 2.25e19T^{2} \)
43 \( 1 - 6.27e9T + 3.99e19T^{2} \)
47 \( 1 - 4.76e9iT - 1.16e20T^{2} \)
53 \( 1 + 2.32e10iT - 4.91e20T^{2} \)
59 \( 1 - 1.03e10iT - 1.77e21T^{2} \)
61 \( 1 - 4.64e10T + 2.65e21T^{2} \)
67 \( 1 + 3.54e10T + 8.18e21T^{2} \)
71 \( 1 - 2.45e11iT - 1.64e22T^{2} \)
73 \( 1 - 2.38e11T + 2.29e22T^{2} \)
79 \( 1 - 3.79e10T + 5.90e22T^{2} \)
83 \( 1 + 2.26e11iT - 1.06e23T^{2} \)
89 \( 1 + 8.22e11iT - 2.46e23T^{2} \)
97 \( 1 - 3.60e11T + 6.93e23T^{2} \)
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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

−13.25756010686103025362988370222, −11.43421665285301068503134129249, −10.98428360714476329738660574261, −9.988948390442688207965127183514, −8.359549683156933934758577715274, −6.42738316366795149689091955584, −6.07582405843986021826280185439, −3.98780048864468071739503091134, −2.94051396956640476046244315843, −0.67834108436230164139687359244, 0.825986681738038834915303966092, 1.76241915453433969668043525945, 4.26951689546699708629594880087, 5.32844297997447631091314304053, 6.61955648931601434033083746399, 8.063130305240283859193427041712, 9.272473240966653346226450619976, 10.71591139338993151393666534140, 12.12730811011925451016328324434, 12.73387777405820445037126362609

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