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Properties

Label 2-483-161.4-c1-0-10
Degree 22
Conductor 483483
Sign 0.109−0.994i0.109 - 0.994i
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  + (−0.957 + 1.34i)2-s + (−0.235 + 0.971i)3-s + (−0.236 − 0.684i)4-s + (0.115 − 2.42i)5-s + (−1.08 − 1.24i)6-s + (2.21 − 1.43i)7-s + (−2.02 − 0.593i)8-s + (−0.888 − 0.458i)9-s + (3.15 + 2.47i)10-s + (−0.243 − 0.341i)11-s + (0.720 − 0.0688i)12-s + (−0.871 + 6.05i)13-s + (−0.189 + 4.36i)14-s + (2.33 + 0.684i)15-s + (3.87 − 3.04i)16-s + (5.69 − 1.09i)17-s + ⋯
L(s)  = 1  + (−0.676 + 0.950i)2-s + (−0.136 + 0.561i)3-s + (−0.118 − 0.342i)4-s + (0.0516 − 1.08i)5-s + (−0.441 − 0.509i)6-s + (0.838 − 0.544i)7-s + (−0.714 − 0.209i)8-s + (−0.296 − 0.152i)9-s + (0.996 + 0.783i)10-s + (−0.0734 − 0.103i)11-s + (0.208 − 0.0198i)12-s + (−0.241 + 1.68i)13-s + (−0.0505 + 1.16i)14-s + (0.601 + 0.176i)15-s + (0.967 − 0.760i)16-s + (1.38 − 0.266i)17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.109−0.994i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.109−0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.109 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.109−0.994i0.109 - 0.994i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(4,⋅)\chi_{483} (4, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 0.109−0.994i)(2,\ 483,\ (\ :1/2),\ 0.109 - 0.994i)

Particular Values

L(1)L(1) ≈\approx 0.781401+0.700147i0.781401 + 0.700147i
L(12)L(\frac12) ≈\approx 0.781401+0.700147i0.781401 + 0.700147i
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+(0.235−0.971i)T 1 + (0.235 - 0.971i)T
7 1+(−2.21+1.43i)T 1 + (-2.21 + 1.43i)T
23 1+(−3.96−2.69i)T 1 + (-3.96 - 2.69i)T
good2 1+(0.957−1.34i)T+(−0.654−1.89i)T2 1 + (0.957 - 1.34i)T + (-0.654 - 1.89i)T^{2}
5 1+(−0.115+2.42i)T+(−4.97−0.475i)T2 1 + (-0.115 + 2.42i)T + (-4.97 - 0.475i)T^{2}
11 1+(0.243+0.341i)T+(−3.59+10.3i)T2 1 + (0.243 + 0.341i)T + (-3.59 + 10.3i)T^{2}
13 1+(0.871−6.05i)T+(−12.4−3.66i)T2 1 + (0.871 - 6.05i)T + (-12.4 - 3.66i)T^{2}
17 1+(−5.69+1.09i)T+(15.7−6.31i)T2 1 + (-5.69 + 1.09i)T + (15.7 - 6.31i)T^{2}
19 1+(−4.77−0.921i)T+(17.6+7.06i)T2 1 + (-4.77 - 0.921i)T + (17.6 + 7.06i)T^{2}
29 1+(0.408+0.471i)T+(−4.12+28.7i)T2 1 + (0.408 + 0.471i)T + (-4.12 + 28.7i)T^{2}
31 1+(5.22−4.98i)T+(1.47−30.9i)T2 1 + (5.22 - 4.98i)T + (1.47 - 30.9i)T^{2}
37 1+(−6.81−3.51i)T+(21.4+30.1i)T2 1 + (-6.81 - 3.51i)T + (21.4 + 30.1i)T^{2}
41 1+(−3.15−2.02i)T+(17.0+37.2i)T2 1 + (-3.15 - 2.02i)T + (17.0 + 37.2i)T^{2}
43 1+(0.780−0.229i)T+(36.1−23.2i)T2 1 + (0.780 - 0.229i)T + (36.1 - 23.2i)T^{2}
47 1+(4.23+7.33i)T+(−23.5+40.7i)T2 1 + (4.23 + 7.33i)T + (-23.5 + 40.7i)T^{2}
53 1+(3.64+1.45i)T+(38.3+36.5i)T2 1 + (3.64 + 1.45i)T + (38.3 + 36.5i)T^{2}
59 1+(2.86+2.25i)T+(13.9+57.3i)T2 1 + (2.86 + 2.25i)T + (13.9 + 57.3i)T^{2}
61 1+(2.90+11.9i)T+(−54.2+27.9i)T2 1 + (2.90 + 11.9i)T + (-54.2 + 27.9i)T^{2}
67 1+(−11.4−1.09i)T+(65.7+12.6i)T2 1 + (-11.4 - 1.09i)T + (65.7 + 12.6i)T^{2}
71 1+(−5.27−11.5i)T+(−46.4+53.6i)T2 1 + (-5.27 - 11.5i)T + (-46.4 + 53.6i)T^{2}
73 1+(1.53+4.42i)T+(−57.3+45.1i)T2 1 + (1.53 + 4.42i)T + (-57.3 + 45.1i)T^{2}
79 1+(−0.449+0.179i)T+(57.1−54.5i)T2 1 + (-0.449 + 0.179i)T + (57.1 - 54.5i)T^{2}
83 1+(7.46−4.79i)T+(34.4−75.4i)T2 1 + (7.46 - 4.79i)T + (34.4 - 75.4i)T^{2}
89 1+(2.05+1.96i)T+(4.23+88.8i)T2 1 + (2.05 + 1.96i)T + (4.23 + 88.8i)T^{2}
97 1+(5.73+3.68i)T+(40.2+88.2i)T2 1 + (5.73 + 3.68i)T + (40.2 + 88.2i)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

−11.27184604331916961326997286925, −9.779448086166188005710658833703, −9.352073465137809024552653018378, −8.461314108104272531335771813883, −7.67444823596726341146995081464, −6.82837052165995885677193917467, −5.45041843230342653961243985808, −4.82203192598509763276371127912, −3.52241455784931394760487013144, −1.21891776612918093923685589536, 1.05768444198315758994839658686, 2.52665922973584617605716023692, 3.17587055800583406042292143618, 5.33616677747538597923861189608, 6.00126973848646170883363623904, 7.48608438646589085991477968520, 7.979389492839746696241565156835, 9.201516145300641696362650749560, 10.12088109119227222338671196145, 10.87140736258822298273548196716

Graph of the ZZ-function along the critical line