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Properties

Label 2-48-3.2-c12-0-9
Degree 22
Conductor 4848
Sign 0.818+0.574i0.818 + 0.574i
Analytic cond. 43.871743.8717
Root an. cond. 6.623576.62357
Motivic weight 1212
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(48s/2 ΓC(s) L(s)=((0.818+0.574i) Λ‾(13−s)\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}
Λ(s)=(48s/2 ΓC(s+6) L(s)=((0.818+0.574i) Λ‾(1−s)\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: 22
Conductor: 4848    =    24⋅32^{4} \cdot 3
Sign: 0.818+0.574i0.818 + 0.574i
Analytic conductor: 43.871743.8717
Root analytic conductor: 6.623576.62357
Motivic weight: 1212
Rational: no
Arithmetic: yes
Character: χ48(17,⋅)\chi_{48} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 48, ( :6), 0.818+0.574i)(2,\ 48,\ (\ :6),\ 0.818 + 0.574i)

Particular Values

L(132)L(\frac{13}{2}) ≈\approx 1.4555421121.455542112
L(12)L(\frac12) ≈\approx 1.4555421121.455542112
L(7)L(7) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(596.+418.i)T 1 + (596. + 418. i)T
good5 1+2.39e4iT−2.44e8T2 1 + 2.39e4iT - 2.44e8T^{2}
7 1+3.21e4T+1.38e10T2 1 + 3.21e4T + 1.38e10T^{2}
11 1−2.66e6iT−3.13e12T2 1 - 2.66e6iT - 3.13e12T^{2}
13 1−7.50e6T+2.32e13T2 1 - 7.50e6T + 2.32e13T^{2}
17 1+1.32e6iT−5.82e14T2 1 + 1.32e6iT - 5.82e14T^{2}
19 1+3.25e7T+2.21e15T2 1 + 3.25e7T + 2.21e15T^{2}
23 1−8.48e7iT−2.19e16T2 1 - 8.48e7iT - 2.19e16T^{2}
29 1−8.40e8iT−3.53e17T2 1 - 8.40e8iT - 3.53e17T^{2}
31 1−1.20e9T+7.87e17T2 1 - 1.20e9T + 7.87e17T^{2}
37 1−1.09e9T+6.58e18T2 1 - 1.09e9T + 6.58e18T^{2}
41 1+5.50e9iT−2.25e19T2 1 + 5.50e9iT - 2.25e19T^{2}
43 1−6.27e9T+3.99e19T2 1 - 6.27e9T + 3.99e19T^{2}
47 1+4.76e9iT−1.16e20T2 1 + 4.76e9iT - 1.16e20T^{2}
53 1−2.32e10iT−4.91e20T2 1 - 2.32e10iT - 4.91e20T^{2}
59 1+1.03e10iT−1.77e21T2 1 + 1.03e10iT - 1.77e21T^{2}
61 1−4.64e10T+2.65e21T2 1 - 4.64e10T + 2.65e21T^{2}
67 1+3.54e10T+8.18e21T2 1 + 3.54e10T + 8.18e21T^{2}
71 1+2.45e11iT−1.64e22T2 1 + 2.45e11iT - 1.64e22T^{2}
73 1−2.38e11T+2.29e22T2 1 - 2.38e11T + 2.29e22T^{2}
79 1−3.79e10T+5.90e22T2 1 - 3.79e10T + 5.90e22T^{2}
83 1−2.26e11iT−1.06e23T2 1 - 2.26e11iT - 1.06e23T^{2}
89 1−8.22e11iT−2.46e23T2 1 - 8.22e11iT - 2.46e23T^{2}
97 1−3.60e11T+6.93e23T2 1 - 3.60e11T + 6.93e23T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

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

Graph of the ZZ-function along the critical line