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Properties

Label 2-483-161.10-c1-0-3
Degree 22
Conductor 483483
Sign 0.481−0.876i0.481 - 0.876i
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  + (0.467 − 1.35i)2-s + (−0.458 − 0.888i)3-s + (−0.0329 − 0.0259i)4-s + (−3.87 + 0.369i)5-s + (−1.41 + 0.203i)6-s + (−0.818 + 2.51i)7-s + (2.35 − 1.51i)8-s + (−0.580 + 0.814i)9-s + (−1.31 + 5.40i)10-s + (−2.90 + 1.00i)11-s + (−0.00793 + 0.0411i)12-s + (1.82 + 6.21i)13-s + (3.01 + 2.28i)14-s + (2.10 + 3.27i)15-s + (−0.962 − 3.96i)16-s + (−5.39 − 2.15i)17-s + ⋯
L(s)  = 1  + (0.330 − 0.954i)2-s + (−0.264 − 0.513i)3-s + (−0.0164 − 0.0129i)4-s + (−1.73 + 0.165i)5-s + (−0.577 + 0.0830i)6-s + (−0.309 + 0.950i)7-s + (0.832 − 0.534i)8-s + (−0.193 + 0.271i)9-s + (−0.414 + 1.70i)10-s + (−0.876 + 0.303i)11-s + (−0.00229 + 0.0118i)12-s + (0.505 + 1.72i)13-s + (0.805 + 0.609i)14-s + (0.543 + 0.845i)15-s + (−0.240 − 0.991i)16-s + (−1.30 − 0.523i)17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.481−0.876i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.481−0.876i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.481−0.876i0.481 - 0.876i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(10,⋅)\chi_{483} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 0.481−0.876i)(2,\ 483,\ (\ :1/2),\ 0.481 - 0.876i)

Particular Values

L(1)L(1) ≈\approx 0.518750+0.306706i0.518750 + 0.306706i
L(12)L(\frac12) ≈\approx 0.518750+0.306706i0.518750 + 0.306706i
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+(0.458+0.888i)T 1 + (0.458 + 0.888i)T
7 1+(0.818−2.51i)T 1 + (0.818 - 2.51i)T
23 1+(−1.31−4.61i)T 1 + (-1.31 - 4.61i)T
good2 1+(−0.467+1.35i)T+(−1.57−1.23i)T2 1 + (-0.467 + 1.35i)T + (-1.57 - 1.23i)T^{2}
5 1+(3.87−0.369i)T+(4.90−0.946i)T2 1 + (3.87 - 0.369i)T + (4.90 - 0.946i)T^{2}
11 1+(2.90−1.00i)T+(8.64−6.79i)T2 1 + (2.90 - 1.00i)T + (8.64 - 6.79i)T^{2}
13 1+(−1.82−6.21i)T+(−10.9+7.02i)T2 1 + (-1.82 - 6.21i)T + (-10.9 + 7.02i)T^{2}
17 1+(5.39+2.15i)T+(12.3+11.7i)T2 1 + (5.39 + 2.15i)T + (12.3 + 11.7i)T^{2}
19 1+(−2.04+0.819i)T+(13.7−13.1i)T2 1 + (-2.04 + 0.819i)T + (13.7 - 13.1i)T^{2}
29 1+(−0.917−6.38i)T+(−27.8+8.17i)T2 1 + (-0.917 - 6.38i)T + (-27.8 + 8.17i)T^{2}
31 1+(9.29−0.442i)T+(30.8−2.94i)T2 1 + (9.29 - 0.442i)T + (30.8 - 2.94i)T^{2}
37 1+(5.57+3.96i)T+(12.1+34.9i)T2 1 + (5.57 + 3.96i)T + (12.1 + 34.9i)T^{2}
41 1+(−0.0145−0.00663i)T+(26.8+30.9i)T2 1 + (-0.0145 - 0.00663i)T + (26.8 + 30.9i)T^{2}
43 1+(−1.64+2.56i)T+(−17.8−39.1i)T2 1 + (-1.64 + 2.56i)T + (-17.8 - 39.1i)T^{2}
47 1+(3.20−1.85i)T+(23.5−40.7i)T2 1 + (3.20 - 1.85i)T + (23.5 - 40.7i)T^{2}
53 1+(1.78+1.87i)T+(−2.52+52.9i)T2 1 + (1.78 + 1.87i)T + (-2.52 + 52.9i)T^{2}
59 1+(0.821+0.199i)T+(52.4+27.0i)T2 1 + (0.821 + 0.199i)T + (52.4 + 27.0i)T^{2}
61 1+(−6.26−3.23i)T+(35.3+49.6i)T2 1 + (-6.26 - 3.23i)T + (35.3 + 49.6i)T^{2}
67 1+(−0.235−1.22i)T+(−62.2+24.9i)T2 1 + (-0.235 - 1.22i)T + (-62.2 + 24.9i)T^{2}
71 1+(−1.17−1.35i)T+(−10.1+70.2i)T2 1 + (-1.17 - 1.35i)T + (-10.1 + 70.2i)T^{2}
73 1+(1.50−1.91i)T+(−17.2−70.9i)T2 1 + (1.50 - 1.91i)T + (-17.2 - 70.9i)T^{2}
79 1+(10.5−11.0i)T+(−3.75−78.9i)T2 1 + (10.5 - 11.0i)T + (-3.75 - 78.9i)T^{2}
83 1+(−0.229−0.502i)T+(−54.3+62.7i)T2 1 + (-0.229 - 0.502i)T + (-54.3 + 62.7i)T^{2}
89 1+(0.376−7.90i)T+(−88.5−8.45i)T2 1 + (0.376 - 7.90i)T + (-88.5 - 8.45i)T^{2}
97 1+(1.60−3.50i)T+(−63.5−73.3i)T2 1 + (1.60 - 3.50i)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

−11.27195748759274903425823873486, −10.94002967478718596286700715042, −9.327487725060606333788088149163, −8.475665811825993517921143534867, −7.15818051612462505995352100283, −6.99912825112562251972599961193, −5.17182625781909227665028431960, −4.10456371122198942512385964960, −3.12907083416560823447975912529, −1.93919564458838263201491268173, 0.32699724796012134577727293530, 3.27391597963289787878958312810, 4.23319671687445051823193336827, 5.08362534715877011594277169351, 6.19172359162595274456886667906, 7.25543636562923207489975371948, 7.929211134793409495910474869372, 8.553761648413202509494115913335, 10.35019817365535013771588237105, 10.78860894493311386193142177005

Graph of the ZZ-function along the critical line