↑

Properties

Label 2-483-483.41-c1-0-6
Degree 22
Conductor 483483
Sign 0.0800+0.996i0.0800 + 0.996i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.601 + 2.04i)2-s + (−0.160 + 1.72i)3-s + (−2.14 − 1.38i)4-s + (−0.358 + 2.49i)5-s + (−3.43 − 1.36i)6-s + (−2.54 + 0.719i)7-s + (0.896 − 0.776i)8-s + (−2.94 − 0.552i)9-s + (−4.88 − 2.23i)10-s + (0.357 + 1.21i)11-s + (2.72 − 3.48i)12-s + (6.26 + 2.86i)13-s + (0.0574 − 5.64i)14-s + (−4.24 − 1.01i)15-s + (−1.07 − 2.34i)16-s + (4.94 − 3.18i)17-s + ⋯
L(s)  = 1  + (−0.425 + 1.44i)2-s + (−0.0924 + 0.995i)3-s + (−1.07 − 0.690i)4-s + (−0.160 + 1.11i)5-s + (−1.40 − 0.557i)6-s + (−0.962 + 0.271i)7-s + (0.316 − 0.274i)8-s + (−0.982 − 0.184i)9-s + (−1.54 − 0.706i)10-s + (0.107 + 0.366i)11-s + (0.787 − 1.00i)12-s + (1.73 + 0.793i)13-s + (0.0153 − 1.50i)14-s + (−1.09 − 0.262i)15-s + (−0.267 − 0.586i)16-s + (1.20 − 0.771i)17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.0800+0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0800 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.0800+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0800 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.0800+0.996i0.0800 + 0.996i
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), 0.0800+0.996i)(2,\ 483,\ (\ :1/2),\ 0.0800 + 0.996i)

Particular Values

L(1)L(1) ≈\approx 0.550514−0.508063i0.550514 - 0.508063i
L(12)L(\frac12) ≈\approx 0.550514−0.508063i0.550514 - 0.508063i
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+(0.160−1.72i)T 1 + (0.160 - 1.72i)T
7 1+(2.54−0.719i)T 1 + (2.54 - 0.719i)T
23 1+(3.20+3.57i)T 1 + (3.20 + 3.57i)T
good2 1+(0.601−2.04i)T+(−1.68−1.08i)T2 1 + (0.601 - 2.04i)T + (-1.68 - 1.08i)T^{2}
5 1+(0.358−2.49i)T+(−4.79−1.40i)T2 1 + (0.358 - 2.49i)T + (-4.79 - 1.40i)T^{2}
11 1+(−0.357−1.21i)T+(−9.25+5.94i)T2 1 + (-0.357 - 1.21i)T + (-9.25 + 5.94i)T^{2}
13 1+(−6.26−2.86i)T+(8.51+9.82i)T2 1 + (-6.26 - 2.86i)T + (8.51 + 9.82i)T^{2}
17 1+(−4.94+3.18i)T+(7.06−15.4i)T2 1 + (-4.94 + 3.18i)T + (7.06 - 15.4i)T^{2}
19 1+(2.59−4.03i)T+(−7.89−17.2i)T2 1 + (2.59 - 4.03i)T + (-7.89 - 17.2i)T^{2}
29 1+(−2.68−4.17i)T+(−12.0+26.3i)T2 1 + (-2.68 - 4.17i)T + (-12.0 + 26.3i)T^{2}
31 1+(1.68−1.46i)T+(4.41−30.6i)T2 1 + (1.68 - 1.46i)T + (4.41 - 30.6i)T^{2}
37 1+(0.484+3.37i)T+(−35.5+10.4i)T2 1 + (0.484 + 3.37i)T + (-35.5 + 10.4i)T^{2}
41 1+(0.362−2.52i)T+(−39.3−11.5i)T2 1 + (0.362 - 2.52i)T + (-39.3 - 11.5i)T^{2}
43 1+(−0.810+0.935i)T+(−6.11−42.5i)T2 1 + (-0.810 + 0.935i)T + (-6.11 - 42.5i)T^{2}
47 1+4.02T+47T2 1 + 4.02T + 47T^{2}
53 1+(5.33−2.43i)T+(34.7−40.0i)T2 1 + (5.33 - 2.43i)T + (34.7 - 40.0i)T^{2}
59 1+(1.15−2.52i)T+(−38.6−44.5i)T2 1 + (1.15 - 2.52i)T + (-38.6 - 44.5i)T^{2}
61 1+(−2.90+2.51i)T+(8.68−60.3i)T2 1 + (-2.90 + 2.51i)T + (8.68 - 60.3i)T^{2}
67 1+(−10.4−3.07i)T+(56.3+36.2i)T2 1 + (-10.4 - 3.07i)T + (56.3 + 36.2i)T^{2}
71 1+(−4.05+13.8i)T+(−59.7−38.3i)T2 1 + (-4.05 + 13.8i)T + (-59.7 - 38.3i)T^{2}
73 1+(8.37−13.0i)T+(−30.3−66.4i)T2 1 + (8.37 - 13.0i)T + (-30.3 - 66.4i)T^{2}
79 1+(1.68−3.68i)T+(−51.7−59.7i)T2 1 + (1.68 - 3.68i)T + (-51.7 - 59.7i)T^{2}
83 1+(0.734+5.10i)T+(−79.6+23.3i)T2 1 + (0.734 + 5.10i)T + (-79.6 + 23.3i)T^{2}
89 1+(8.18−9.44i)T+(−12.6−88.0i)T2 1 + (8.18 - 9.44i)T + (-12.6 - 88.0i)T^{2}
97 1+(10.8+1.56i)T+(93.0+27.3i)T2 1 + (10.8 + 1.56i)T + (93.0 + 27.3i)T^{2}
show more
show less
   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

−11.39538114726398853757968438002, −10.47062008102648934216900678864, −9.720649286306336649783117932529, −8.901375085689388451212813204024, −8.088577904620492956553109123246, −6.81965529583992685549360807401, −6.32224383309887311125634385626, −5.50028884090722291614366795648, −3.99547395841661776382812241759, −3.02899231875042393883519878830, 0.57212326680893242573649941484, 1.45109033065118518363308535829, 3.05452783817681253912388120132, 3.91053282994385218359053107275, 5.68166759285141247660612057512, 6.47632989165145711588869107392, 8.091742016149297796143615899557, 8.540904598484701760388643164237, 9.483643178774603888903850626001, 10.46781952803526575331894080860

Graph of the ZZ-function along the critical line