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Properties

Label 2-483-23.13-c1-0-8
Degree 22
Conductor 483483
Sign −0.888−0.458i-0.888 - 0.458i
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  + (1.64 + 1.89i)2-s + (0.841 + 0.540i)3-s + (−0.614 + 4.27i)4-s + (−0.415 + 0.909i)5-s + (0.357 + 2.48i)6-s + (−0.959 − 0.281i)7-s + (−4.89 + 3.14i)8-s + (0.415 + 0.909i)9-s + (−2.41 + 0.708i)10-s + (0.0756 − 0.0872i)11-s + (−2.82 + 3.26i)12-s + (1.24 − 0.366i)13-s + (−1.04 − 2.28i)14-s + (−0.841 + 0.540i)15-s + (−5.75 − 1.69i)16-s + (−0.556 − 3.87i)17-s + ⋯
L(s)  = 1  + (1.16 + 1.34i)2-s + (0.485 + 0.312i)3-s + (−0.307 + 2.13i)4-s + (−0.185 + 0.406i)5-s + (0.146 + 1.01i)6-s + (−0.362 − 0.106i)7-s + (−1.73 + 1.11i)8-s + (0.138 + 0.303i)9-s + (−0.762 + 0.223i)10-s + (0.0227 − 0.0263i)11-s + (−0.816 + 0.941i)12-s + (0.345 − 0.101i)13-s + (−0.279 − 0.611i)14-s + (−0.217 + 0.139i)15-s + (−1.43 − 0.422i)16-s + (−0.135 − 0.939i)17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((−0.888−0.458i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((−0.888−0.458i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: −0.888−0.458i-0.888 - 0.458i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(358,⋅)\chi_{483} (358, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), −0.888−0.458i)(2,\ 483,\ (\ :1/2),\ -0.888 - 0.458i)

Particular Values

L(1)L(1) ≈\approx 0.644451+2.65339i0.644451 + 2.65339i
L(12)L(\frac12) ≈\approx 0.644451+2.65339i0.644451 + 2.65339i
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+(−0.841−0.540i)T 1 + (-0.841 - 0.540i)T
7 1+(0.959+0.281i)T 1 + (0.959 + 0.281i)T
23 1+(−4.67+1.07i)T 1 + (-4.67 + 1.07i)T
good2 1+(−1.64−1.89i)T+(−0.284+1.97i)T2 1 + (-1.64 - 1.89i)T + (-0.284 + 1.97i)T^{2}
5 1+(0.415−0.909i)T+(−3.27−3.77i)T2 1 + (0.415 - 0.909i)T + (-3.27 - 3.77i)T^{2}
11 1+(−0.0756+0.0872i)T+(−1.56−10.8i)T2 1 + (-0.0756 + 0.0872i)T + (-1.56 - 10.8i)T^{2}
13 1+(−1.24+0.366i)T+(10.9−7.02i)T2 1 + (-1.24 + 0.366i)T + (10.9 - 7.02i)T^{2}
17 1+(0.556+3.87i)T+(−16.3+4.78i)T2 1 + (0.556 + 3.87i)T + (-16.3 + 4.78i)T^{2}
19 1+(−0.195+1.35i)T+(−18.2−5.35i)T2 1 + (-0.195 + 1.35i)T + (-18.2 - 5.35i)T^{2}
29 1+(−0.291−2.02i)T+(−27.8+8.17i)T2 1 + (-0.291 - 2.02i)T + (-27.8 + 8.17i)T^{2}
31 1+(4.96−3.19i)T+(12.8−28.1i)T2 1 + (4.96 - 3.19i)T + (12.8 - 28.1i)T^{2}
37 1+(−0.215−0.471i)T+(−24.2+27.9i)T2 1 + (-0.215 - 0.471i)T + (-24.2 + 27.9i)T^{2}
41 1+(−3.90+8.55i)T+(−26.8−30.9i)T2 1 + (-3.90 + 8.55i)T + (-26.8 - 30.9i)T^{2}
43 1+(5.54+3.56i)T+(17.8+39.1i)T2 1 + (5.54 + 3.56i)T + (17.8 + 39.1i)T^{2}
47 1−5.90T+47T2 1 - 5.90T + 47T^{2}
53 1+(4.20+1.23i)T+(44.5+28.6i)T2 1 + (4.20 + 1.23i)T + (44.5 + 28.6i)T^{2}
59 1+(−3.46+1.01i)T+(49.6−31.8i)T2 1 + (-3.46 + 1.01i)T + (49.6 - 31.8i)T^{2}
61 1+(−4.23+2.72i)T+(25.3−55.4i)T2 1 + (-4.23 + 2.72i)T + (25.3 - 55.4i)T^{2}
67 1+(−9.18−10.5i)T+(−9.53+66.3i)T2 1 + (-9.18 - 10.5i)T + (-9.53 + 66.3i)T^{2}
71 1+(6.24+7.20i)T+(−10.1+70.2i)T2 1 + (6.24 + 7.20i)T + (-10.1 + 70.2i)T^{2}
73 1+(0.0848−0.590i)T+(−70.0−20.5i)T2 1 + (0.0848 - 0.590i)T + (-70.0 - 20.5i)T^{2}
79 1+(9.06−2.66i)T+(66.4−42.7i)T2 1 + (9.06 - 2.66i)T + (66.4 - 42.7i)T^{2}
83 1+(5.33+11.6i)T+(−54.3+62.7i)T2 1 + (5.33 + 11.6i)T + (-54.3 + 62.7i)T^{2}
89 1+(4.28+2.75i)T+(36.9+80.9i)T2 1 + (4.28 + 2.75i)T + (36.9 + 80.9i)T^{2}
97 1+(−3.21+7.03i)T+(−63.5−73.3i)T2 1 + (-3.21 + 7.03i)T + (-63.5 - 73.3i)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

−11.52108904210203568349474427926, −10.54040174542796452726395071789, −9.194895400150449239652103994243, −8.489632817428618238233824663622, −7.15663122475639869555850805231, −7.03050216703305691811793138804, −5.65865074656576041291610233561, −4.82311772426620951149728581797, −3.70403804739330526177862885613, −2.91941146309359990949652329150, 1.26944474738682491758459565163, 2.54912812532898293153801710450, 3.60336848174369651249963354214, 4.44553460926002820326401354246, 5.61273391784012912748947645690, 6.58733794157766764080933635699, 8.043946748518616238266553753401, 9.084858239526924093975496637547, 9.932380890556536568883624531516, 10.88638362474329567175981327573

Graph of the ZZ-function along the critical line