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Properties

Label 2-483-483.377-c1-0-36
Degree 22
Conductor 483483
Sign 0.977−0.210i0.977 - 0.210i
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  + (0.454 + 1.54i)2-s + (−0.712 + 1.57i)3-s + (−0.504 + 0.323i)4-s + (−0.562 − 3.91i)5-s + (−2.76 − 0.385i)6-s + (−1.48 − 2.18i)7-s + (1.70 + 1.47i)8-s + (−1.98 − 2.25i)9-s + (5.79 − 2.64i)10-s + (0.644 − 2.19i)11-s + (−0.152 − 1.02i)12-s + (1.76 − 0.807i)13-s + (2.70 − 3.29i)14-s + (6.57 + 1.90i)15-s + (−2.01 + 4.40i)16-s + (6.25 + 4.01i)17-s + ⋯
L(s)  = 1  + (0.321 + 1.09i)2-s + (−0.411 + 0.911i)3-s + (−0.252 + 0.161i)4-s + (−0.251 − 1.74i)5-s + (−1.12 − 0.157i)6-s + (−0.562 − 0.826i)7-s + (0.603 + 0.522i)8-s + (−0.661 − 0.750i)9-s + (1.83 − 0.836i)10-s + (0.194 − 0.662i)11-s + (−0.0438 − 0.296i)12-s + (0.490 − 0.224i)13-s + (0.723 − 0.880i)14-s + (1.69 + 0.490i)15-s + (−0.502 + 1.10i)16-s + (1.51 + 0.974i)17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.977−0.210i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.977−0.210i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.977−0.210i0.977 - 0.210i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(377,⋅)\chi_{483} (377, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 0.977−0.210i)(2,\ 483,\ (\ :1/2),\ 0.977 - 0.210i)

Particular Values

L(1)L(1) ≈\approx 1.38463+0.147041i1.38463 + 0.147041i
L(12)L(\frac12) ≈\approx 1.38463+0.147041i1.38463 + 0.147041i
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+(0.712−1.57i)T 1 + (0.712 - 1.57i)T
7 1+(1.48+2.18i)T 1 + (1.48 + 2.18i)T
23 1+(−3.42+3.35i)T 1 + (-3.42 + 3.35i)T
good2 1+(−0.454−1.54i)T+(−1.68+1.08i)T2 1 + (-0.454 - 1.54i)T + (-1.68 + 1.08i)T^{2}
5 1+(0.562+3.91i)T+(−4.79+1.40i)T2 1 + (0.562 + 3.91i)T + (-4.79 + 1.40i)T^{2}
11 1+(−0.644+2.19i)T+(−9.25−5.94i)T2 1 + (-0.644 + 2.19i)T + (-9.25 - 5.94i)T^{2}
13 1+(−1.76+0.807i)T+(8.51−9.82i)T2 1 + (-1.76 + 0.807i)T + (8.51 - 9.82i)T^{2}
17 1+(−6.25−4.01i)T+(7.06+15.4i)T2 1 + (-6.25 - 4.01i)T + (7.06 + 15.4i)T^{2}
19 1+(2.11+3.29i)T+(−7.89+17.2i)T2 1 + (2.11 + 3.29i)T + (-7.89 + 17.2i)T^{2}
29 1+(−0.791+1.23i)T+(−12.0−26.3i)T2 1 + (-0.791 + 1.23i)T + (-12.0 - 26.3i)T^{2}
31 1+(4.87+4.22i)T+(4.41+30.6i)T2 1 + (4.87 + 4.22i)T + (4.41 + 30.6i)T^{2}
37 1+(−0.362+2.52i)T+(−35.5−10.4i)T2 1 + (-0.362 + 2.52i)T + (-35.5 - 10.4i)T^{2}
41 1+(0.875+6.09i)T+(−39.3+11.5i)T2 1 + (0.875 + 6.09i)T + (-39.3 + 11.5i)T^{2}
43 1+(2.49+2.87i)T+(−6.11+42.5i)T2 1 + (2.49 + 2.87i)T + (-6.11 + 42.5i)T^{2}
47 1+2.63T+47T2 1 + 2.63T + 47T^{2}
53 1+(−4.87−2.22i)T+(34.7+40.0i)T2 1 + (-4.87 - 2.22i)T + (34.7 + 40.0i)T^{2}
59 1+(−5.91−12.9i)T+(−38.6+44.5i)T2 1 + (-5.91 - 12.9i)T + (-38.6 + 44.5i)T^{2}
61 1+(−10.9−9.44i)T+(8.68+60.3i)T2 1 + (-10.9 - 9.44i)T + (8.68 + 60.3i)T^{2}
67 1+(8.41−2.47i)T+(56.3−36.2i)T2 1 + (8.41 - 2.47i)T + (56.3 - 36.2i)T^{2}
71 1+(1.65+5.63i)T+(−59.7+38.3i)T2 1 + (1.65 + 5.63i)T + (-59.7 + 38.3i)T^{2}
73 1+(−0.871−1.35i)T+(−30.3+66.4i)T2 1 + (-0.871 - 1.35i)T + (-30.3 + 66.4i)T^{2}
79 1+(5.26+11.5i)T+(−51.7+59.7i)T2 1 + (5.26 + 11.5i)T + (-51.7 + 59.7i)T^{2}
83 1+(0.586−4.08i)T+(−79.6−23.3i)T2 1 + (0.586 - 4.08i)T + (-79.6 - 23.3i)T^{2}
89 1+(−4.01−4.63i)T+(−12.6+88.0i)T2 1 + (-4.01 - 4.63i)T + (-12.6 + 88.0i)T^{2}
97 1+(3.81−0.548i)T+(93.0−27.3i)T2 1 + (3.81 - 0.548i)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

−10.88468916917657272665483115274, −10.13510344164238024987902900648, −8.907369644540864211944603510030, −8.434865179591303363016274426873, −7.28836685311345267388737832372, −5.99053163963785821023661407035, −5.50194243269742555947712199501, −4.45251020083104027921073398709, −3.74028000329474507291025362459, −0.853666150100695050026034439233, 1.75763006604654875623251681662, 2.88729264771537048153698554718, 3.48183593956013591863588864217, 5.33982733907144747880269257642, 6.56105399451251611224085173533, 7.05141153047334685578918090506, 7.987025830808499595266239418697, 9.649811895274992038081240974201, 10.32598074084948949212206027352, 11.32710522283615484302862769964

Graph of the ZZ-function along the critical line