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Properties

Label 2-483-7.4-c1-0-28
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  + (−1.25 − 2.17i)2-s + (0.5 − 0.866i)3-s + (−2.16 + 3.74i)4-s + (−2.15 − 3.73i)5-s − 2.51·6-s + (1.42 − 2.22i)7-s + 5.85·8-s + (−0.499 − 0.866i)9-s + (−5.42 + 9.40i)10-s + (−0.864 + 1.49i)11-s + (2.16 + 3.74i)12-s − 2.88·13-s + (−6.64 − 0.306i)14-s − 4.31·15-s + (−3.03 − 5.25i)16-s + (2.46 − 4.27i)17-s + ⋯
L(s)  = 1  + (−0.889 − 1.54i)2-s + (0.288 − 0.499i)3-s + (−1.08 + 1.87i)4-s + (−0.965 − 1.67i)5-s − 1.02·6-s + (0.539 − 0.842i)7-s + 2.06·8-s + (−0.166 − 0.288i)9-s + (−1.71 + 2.97i)10-s + (−0.260 + 0.451i)11-s + (0.624 + 1.08i)12-s − 0.801·13-s + (−1.77 − 0.0818i)14-s − 1.11·15-s + (−0.757 − 1.31i)16-s + (0.598 − 1.03i)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(277,⋅)\chi_{483} (277, \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 0.412941+0.370060i0.412941 + 0.370060i
L(12)L(\frac12) ≈\approx 0.412941+0.370060i0.412941 + 0.370060i
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1+(−1.42+2.22i)T 1 + (-1.42 + 2.22i)T
23 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good2 1+(1.25+2.17i)T+(−1+1.73i)T2 1 + (1.25 + 2.17i)T + (-1 + 1.73i)T^{2}
5 1+(2.15+3.73i)T+(−2.5+4.33i)T2 1 + (2.15 + 3.73i)T + (-2.5 + 4.33i)T^{2}
11 1+(0.864−1.49i)T+(−5.5−9.52i)T2 1 + (0.864 - 1.49i)T + (-5.5 - 9.52i)T^{2}
13 1+2.88T+13T2 1 + 2.88T + 13T^{2}
17 1+(−2.46+4.27i)T+(−8.5−14.7i)T2 1 + (-2.46 + 4.27i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.22+2.11i)T+(−9.5+16.4i)T2 1 + (1.22 + 2.11i)T + (-9.5 + 16.4i)T^{2}
29 1−8.05T+29T2 1 - 8.05T + 29T^{2}
31 1+(0.457−0.793i)T+(−15.5−26.8i)T2 1 + (0.457 - 0.793i)T + (-15.5 - 26.8i)T^{2}
37 1+(−4.57−7.92i)T+(−18.5+32.0i)T2 1 + (-4.57 - 7.92i)T + (-18.5 + 32.0i)T^{2}
41 1+6.60T+41T2 1 + 6.60T + 41T^{2}
43 1−10.7T+43T2 1 - 10.7T + 43T^{2}
47 1+(−0.199−0.346i)T+(−23.5+40.7i)T2 1 + (-0.199 - 0.346i)T + (-23.5 + 40.7i)T^{2}
53 1+(3.27−5.66i)T+(−26.5−45.8i)T2 1 + (3.27 - 5.66i)T + (-26.5 - 45.8i)T^{2}
59 1+(−0.917+1.58i)T+(−29.5−51.0i)T2 1 + (-0.917 + 1.58i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.650−1.12i)T+(−30.5+52.8i)T2 1 + (-0.650 - 1.12i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.10+8.84i)T+(−33.5−58.0i)T2 1 + (-5.10 + 8.84i)T + (-33.5 - 58.0i)T^{2}
71 1+9.00T+71T2 1 + 9.00T + 71T^{2}
73 1+(−1.30+2.26i)T+(−36.5−63.2i)T2 1 + (-1.30 + 2.26i)T + (-36.5 - 63.2i)T^{2}
79 1+(5.09+8.82i)T+(−39.5+68.4i)T2 1 + (5.09 + 8.82i)T + (-39.5 + 68.4i)T^{2}
83 1+10.8T+83T2 1 + 10.8T + 83T^{2}
89 1+(7.49+12.9i)T+(−44.5+77.0i)T2 1 + (7.49 + 12.9i)T + (-44.5 + 77.0i)T^{2}
97 1+3.07T+97T2 1 + 3.07T + 97T^{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

−10.23995245316993649429289056912, −9.437537632847720633580778680197, −8.600825157631603425423167916697, −7.928674909324020828784319672635, −7.31913364558359978795307794347, −4.85355418735337844085594291372, −4.31496485695622351621984208505, −2.91775488674725555285131155745, −1.41461360840507551873904449931, −0.47379162985919674995987383752, 2.67268366338395328094695585087, 4.10326465661217841929067795252, 5.50250125500398025538565398791, 6.32673060117596687740826745553, 7.34027656784809240481684266745, 8.031249217391877301365620630881, 8.552160056077467560900164965607, 9.819165805263082626560846447424, 10.45415955927797990984855748834, 11.27752930333724454763148185898

Graph of the ZZ-function along the critical line