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Properties

Label 2-483-483.41-c1-0-1
Degree 22
Conductor 483483
Sign −0.513+0.857i-0.513 + 0.857i
Analytic cond. 3.856773.85677
Root an. cond. 1.963861.96386
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.493 + 1.68i)2-s + (0.151 − 1.72i)3-s + (−0.903 − 0.580i)4-s + (−0.558 + 3.88i)5-s + (2.82 + 1.10i)6-s + (−1.99 − 1.74i)7-s + (−1.22 + 1.06i)8-s + (−2.95 − 0.524i)9-s + (−6.26 − 2.86i)10-s + (−1.22 − 4.17i)11-s + (−1.13 + 1.47i)12-s + (−1.88 − 0.861i)13-s + (3.91 − 2.49i)14-s + (6.62 + 1.55i)15-s + (−2.07 − 4.54i)16-s + (−1.78 + 1.14i)17-s + ⋯
L(s)  = 1  + (−0.349 + 1.18i)2-s + (0.0877 − 0.996i)3-s + (−0.451 − 0.290i)4-s + (−0.249 + 1.73i)5-s + (1.15 + 0.452i)6-s + (−0.753 − 0.657i)7-s + (−0.433 + 0.375i)8-s + (−0.984 − 0.174i)9-s + (−1.98 − 0.904i)10-s + (−0.370 − 1.26i)11-s + (−0.328 + 0.424i)12-s + (−0.523 − 0.238i)13-s + (1.04 − 0.666i)14-s + (1.70 + 0.401i)15-s + (−0.518 − 1.13i)16-s + (−0.431 + 0.277i)17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((−0.513+0.857i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((−0.513+0.857i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: −0.513+0.857i-0.513 + 0.857i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(41,⋅)\chi_{483} (41, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), −0.513+0.857i)(2,\ 483,\ (\ :1/2),\ -0.513 + 0.857i)

Particular Values

L(1)L(1) ≈\approx 0.0503626−0.0888679i0.0503626 - 0.0888679i
L(12)L(\frac12) ≈\approx 0.0503626−0.0888679i0.0503626 - 0.0888679i
L(32)L(\frac{3}{2}) 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)
bad3 1+(−0.151+1.72i)T 1 + (-0.151 + 1.72i)T
7 1+(1.99+1.74i)T 1 + (1.99 + 1.74i)T
23 1+(−4.27−2.17i)T 1 + (-4.27 - 2.17i)T
good2 1+(0.493−1.68i)T+(−1.68−1.08i)T2 1 + (0.493 - 1.68i)T + (-1.68 - 1.08i)T^{2}
5 1+(0.558−3.88i)T+(−4.79−1.40i)T2 1 + (0.558 - 3.88i)T + (-4.79 - 1.40i)T^{2}
11 1+(1.22+4.17i)T+(−9.25+5.94i)T2 1 + (1.22 + 4.17i)T + (-9.25 + 5.94i)T^{2}
13 1+(1.88+0.861i)T+(8.51+9.82i)T2 1 + (1.88 + 0.861i)T + (8.51 + 9.82i)T^{2}
17 1+(1.78−1.14i)T+(7.06−15.4i)T2 1 + (1.78 - 1.14i)T + (7.06 - 15.4i)T^{2}
19 1+(1.42−2.21i)T+(−7.89−17.2i)T2 1 + (1.42 - 2.21i)T + (-7.89 - 17.2i)T^{2}
29 1+(0.621+0.967i)T+(−12.0+26.3i)T2 1 + (0.621 + 0.967i)T + (-12.0 + 26.3i)T^{2}
31 1+(−0.602+0.522i)T+(4.41−30.6i)T2 1 + (-0.602 + 0.522i)T + (4.41 - 30.6i)T^{2}
37 1+(1.02+7.10i)T+(−35.5+10.4i)T2 1 + (1.02 + 7.10i)T + (-35.5 + 10.4i)T^{2}
41 1+(−0.650+4.52i)T+(−39.3−11.5i)T2 1 + (-0.650 + 4.52i)T + (-39.3 - 11.5i)T^{2}
43 1+(5.13−5.92i)T+(−6.11−42.5i)T2 1 + (5.13 - 5.92i)T + (-6.11 - 42.5i)T^{2}
47 1+9.37T+47T2 1 + 9.37T + 47T^{2}
53 1+(8.95−4.09i)T+(34.7−40.0i)T2 1 + (8.95 - 4.09i)T + (34.7 - 40.0i)T^{2}
59 1+(4.97−10.8i)T+(−38.6−44.5i)T2 1 + (4.97 - 10.8i)T + (-38.6 - 44.5i)T^{2}
61 1+(−3.12+2.70i)T+(8.68−60.3i)T2 1 + (-3.12 + 2.70i)T + (8.68 - 60.3i)T^{2}
67 1+(−7.89−2.31i)T+(56.3+36.2i)T2 1 + (-7.89 - 2.31i)T + (56.3 + 36.2i)T^{2}
71 1+(0.509−1.73i)T+(−59.7−38.3i)T2 1 + (0.509 - 1.73i)T + (-59.7 - 38.3i)T^{2}
73 1+(−0.202+0.315i)T+(−30.3−66.4i)T2 1 + (-0.202 + 0.315i)T + (-30.3 - 66.4i)T^{2}
79 1+(0.623−1.36i)T+(−51.7−59.7i)T2 1 + (0.623 - 1.36i)T + (-51.7 - 59.7i)T^{2}
83 1+(−2.36−16.4i)T+(−79.6+23.3i)T2 1 + (-2.36 - 16.4i)T + (-79.6 + 23.3i)T^{2}
89 1+(6.73−7.77i)T+(−12.6−88.0i)T2 1 + (6.73 - 7.77i)T + (-12.6 - 88.0i)T^{2}
97 1+(1.26+0.182i)T+(93.0+27.3i)T2 1 + (1.26 + 0.182i)T + (93.0 + 27.3i)T^{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

−11.28239200519659798773044851898, −10.87839383967539603201968598384, −9.644969246051756436912448720837, −8.397677797627814854662356485027, −7.64920400506735186972109250030, −7.01061216051302074239400446007, −6.41073048922171500228664860127, −5.71519187460177306186477097874, −3.42139923512772847328604392599, −2.67723088902853554840218865686, 0.06257010636379887516061846477, 2.02307061811861521689251343695, 3.23535922143789958958649179397, 4.60639203133252354448512060122, 5.03703852792355663157713340280, 6.58212324082627300284781246906, 8.291564518868112199211996378980, 9.057902641490724392496902157052, 9.561891949919121178939811429022, 10.14265566434746534427957774280

Graph of the ZZ-function along the critical line