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Properties

Label 2-483-23.13-c1-0-3
Degree 22
Conductor 483483
Sign 0.432−0.901i0.432 - 0.901i
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  + (−0.544 − 0.627i)2-s + (0.841 + 0.540i)3-s + (0.186 − 1.29i)4-s + (−1.36 + 2.98i)5-s + (−0.118 − 0.822i)6-s + (−0.959 − 0.281i)7-s + (−2.31 + 1.48i)8-s + (0.415 + 0.909i)9-s + (2.62 − 0.769i)10-s + (−0.745 + 0.860i)11-s + (0.857 − 0.989i)12-s + (4.40 − 1.29i)13-s + (0.345 + 0.755i)14-s + (−2.76 + 1.77i)15-s + (−0.321 − 0.0943i)16-s + (0.756 + 5.26i)17-s + ⋯
L(s)  = 1  + (−0.384 − 0.443i)2-s + (0.485 + 0.312i)3-s + (0.0931 − 0.648i)4-s + (−0.610 + 1.33i)5-s + (−0.0482 − 0.335i)6-s + (−0.362 − 0.106i)7-s + (−0.817 + 0.525i)8-s + (0.138 + 0.303i)9-s + (0.828 − 0.243i)10-s + (−0.224 + 0.259i)11-s + (0.247 − 0.285i)12-s + (1.22 − 0.358i)13-s + (0.0922 + 0.201i)14-s + (−0.713 + 0.458i)15-s + (−0.0803 − 0.0235i)16-s + (0.183 + 1.27i)17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.432−0.901i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.432−0.901i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.432−0.901i0.432 - 0.901i
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), 0.432−0.901i)(2,\ 483,\ (\ :1/2),\ 0.432 - 0.901i)

Particular Values

L(1)L(1) ≈\approx 0.858350+0.540123i0.858350 + 0.540123i
L(12)L(\frac12) ≈\approx 0.858350+0.540123i0.858350 + 0.540123i
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+(−1.20−4.64i)T 1 + (-1.20 - 4.64i)T
good2 1+(0.544+0.627i)T+(−0.284+1.97i)T2 1 + (0.544 + 0.627i)T + (-0.284 + 1.97i)T^{2}
5 1+(1.36−2.98i)T+(−3.27−3.77i)T2 1 + (1.36 - 2.98i)T + (-3.27 - 3.77i)T^{2}
11 1+(0.745−0.860i)T+(−1.56−10.8i)T2 1 + (0.745 - 0.860i)T + (-1.56 - 10.8i)T^{2}
13 1+(−4.40+1.29i)T+(10.9−7.02i)T2 1 + (-4.40 + 1.29i)T + (10.9 - 7.02i)T^{2}
17 1+(−0.756−5.26i)T+(−16.3+4.78i)T2 1 + (-0.756 - 5.26i)T + (-16.3 + 4.78i)T^{2}
19 1+(1.11−7.72i)T+(−18.2−5.35i)T2 1 + (1.11 - 7.72i)T + (-18.2 - 5.35i)T^{2}
29 1+(1.10+7.69i)T+(−27.8+8.17i)T2 1 + (1.10 + 7.69i)T + (-27.8 + 8.17i)T^{2}
31 1+(2.20−1.41i)T+(12.8−28.1i)T2 1 + (2.20 - 1.41i)T + (12.8 - 28.1i)T^{2}
37 1+(−1.90−4.16i)T+(−24.2+27.9i)T2 1 + (-1.90 - 4.16i)T + (-24.2 + 27.9i)T^{2}
41 1+(0.343−0.751i)T+(−26.8−30.9i)T2 1 + (0.343 - 0.751i)T + (-26.8 - 30.9i)T^{2}
43 1+(2.23+1.43i)T+(17.8+39.1i)T2 1 + (2.23 + 1.43i)T + (17.8 + 39.1i)T^{2}
47 1−5.46T+47T2 1 - 5.46T + 47T^{2}
53 1+(10.0+2.94i)T+(44.5+28.6i)T2 1 + (10.0 + 2.94i)T + (44.5 + 28.6i)T^{2}
59 1+(−5.91+1.73i)T+(49.6−31.8i)T2 1 + (-5.91 + 1.73i)T + (49.6 - 31.8i)T^{2}
61 1+(−0.526+0.338i)T+(25.3−55.4i)T2 1 + (-0.526 + 0.338i)T + (25.3 - 55.4i)T^{2}
67 1+(1.30+1.50i)T+(−9.53+66.3i)T2 1 + (1.30 + 1.50i)T + (-9.53 + 66.3i)T^{2}
71 1+(−5.90−6.81i)T+(−10.1+70.2i)T2 1 + (-5.90 - 6.81i)T + (-10.1 + 70.2i)T^{2}
73 1+(−1.55+10.8i)T+(−70.0−20.5i)T2 1 + (-1.55 + 10.8i)T + (-70.0 - 20.5i)T^{2}
79 1+(−4.53+1.33i)T+(66.4−42.7i)T2 1 + (-4.53 + 1.33i)T + (66.4 - 42.7i)T^{2}
83 1+(−2.51−5.50i)T+(−54.3+62.7i)T2 1 + (-2.51 - 5.50i)T + (-54.3 + 62.7i)T^{2}
89 1+(11.7+7.53i)T+(36.9+80.9i)T2 1 + (11.7 + 7.53i)T + (36.9 + 80.9i)T^{2}
97 1+(−1.80+3.95i)T+(−63.5−73.3i)T2 1 + (-1.80 + 3.95i)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

−10.89177020517382046493271253338, −10.29141095476870992224871606859, −9.726098446024424305060516391084, −8.433293094291204379211799950277, −7.73542682507660204075754085890, −6.43569084040702878504831021405, −5.76344405402304282653993345326, −3.88772951021429901994292136083, −3.21440532522295016008423610663, −1.79660520109270860891510396148, 0.67151439655837106534373602846, 2.79555401029301748140922355139, 3.93126426230636419228962665343, 5.05171057668661951835738466960, 6.52198209474314928315021108868, 7.32363076131191476800840069681, 8.258335084950806562126433027026, 9.048588709826122230622599906488, 9.137933949179806865451622565703, 11.01128860994437782226260666677

Graph of the ZZ-function along the critical line