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Properties

Label 2-728-104.101-c1-0-27
Degree 22
Conductor 728728
Sign 0.923+0.383i0.923 + 0.383i
Analytic cond. 5.813105.81310
Root an. cond. 2.411032.41103
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  + (0.687 − 1.23i)2-s + (−1.40 + 0.809i)3-s + (−1.05 − 1.69i)4-s − 2.10·5-s + (0.0358 + 2.28i)6-s + (−0.866 − 0.5i)7-s + (−2.82 + 0.132i)8-s + (−0.190 + 0.330i)9-s + (−1.44 + 2.59i)10-s + (2.17 + 3.76i)11-s + (2.85 + 1.52i)12-s + (3.28 + 1.49i)13-s + (−1.21 + 0.726i)14-s + (2.94 − 1.70i)15-s + (−1.77 + 3.58i)16-s + (2.81 − 4.87i)17-s + ⋯
L(s)  = 1  + (0.486 − 0.873i)2-s + (−0.809 + 0.467i)3-s + (−0.526 − 0.849i)4-s − 0.940·5-s + (0.0146 + 0.934i)6-s + (−0.327 − 0.188i)7-s + (−0.998 + 0.0469i)8-s + (−0.0636 + 0.110i)9-s + (−0.457 + 0.821i)10-s + (0.655 + 1.13i)11-s + (0.823 + 0.441i)12-s + (0.909 + 0.414i)13-s + (−0.324 + 0.194i)14-s + (0.760 − 0.439i)15-s + (−0.444 + 0.895i)16-s + (0.682 − 1.18i)17-s + ⋯

Functional equation

Λ(s)=(728s/2 ΓC(s) L(s)=((0.923+0.383i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(728s/2 ΓC(s+1/2) L(s)=((0.923+0.383i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 728728    =    23⋅7⋅132^{3} \cdot 7 \cdot 13
Sign: 0.923+0.383i0.923 + 0.383i
Analytic conductor: 5.813105.81310
Root analytic conductor: 2.411032.41103
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ728(309,⋅)\chi_{728} (309, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 728, ( :1/2), 0.923+0.383i)(2,\ 728,\ (\ :1/2),\ 0.923 + 0.383i)

Particular Values

L(1)L(1) ≈\approx 1.06491−0.212346i1.06491 - 0.212346i
L(12)L(\frac12) ≈\approx 1.06491−0.212346i1.06491 - 0.212346i
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)
bad2 1+(−0.687+1.23i)T 1 + (-0.687 + 1.23i)T
7 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
13 1+(−3.28−1.49i)T 1 + (-3.28 - 1.49i)T
good3 1+(1.40−0.809i)T+(1.5−2.59i)T2 1 + (1.40 - 0.809i)T + (1.5 - 2.59i)T^{2}
5 1+2.10T+5T2 1 + 2.10T + 5T^{2}
11 1+(−2.17−3.76i)T+(−5.5+9.52i)T2 1 + (-2.17 - 3.76i)T + (-5.5 + 9.52i)T^{2}
17 1+(−2.81+4.87i)T+(−8.5−14.7i)T2 1 + (-2.81 + 4.87i)T + (-8.5 - 14.7i)T^{2}
19 1+(−4.13+7.16i)T+(−9.5−16.4i)T2 1 + (-4.13 + 7.16i)T + (-9.5 - 16.4i)T^{2}
23 1+(−3.04−5.26i)T+(−11.5+19.9i)T2 1 + (-3.04 - 5.26i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.05+0.609i)T+(14.5−25.1i)T2 1 + (-1.05 + 0.609i)T + (14.5 - 25.1i)T^{2}
31 1−7.58iT−31T2 1 - 7.58iT - 31T^{2}
37 1+(1.96+3.40i)T+(−18.5+32.0i)T2 1 + (1.96 + 3.40i)T + (-18.5 + 32.0i)T^{2}
41 1+(−3.53+2.04i)T+(20.5−35.5i)T2 1 + (-3.53 + 2.04i)T + (20.5 - 35.5i)T^{2}
43 1+(−7.82−4.52i)T+(21.5+37.2i)T2 1 + (-7.82 - 4.52i)T + (21.5 + 37.2i)T^{2}
47 1−4.60iT−47T2 1 - 4.60iT - 47T^{2}
53 1−7.82iT−53T2 1 - 7.82iT - 53T^{2}
59 1+(3.60−6.23i)T+(−29.5−51.0i)T2 1 + (3.60 - 6.23i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2.27−1.31i)T+(30.5+52.8i)T2 1 + (-2.27 - 1.31i)T + (30.5 + 52.8i)T^{2}
67 1+(−0.990−1.71i)T+(−33.5+58.0i)T2 1 + (-0.990 - 1.71i)T + (-33.5 + 58.0i)T^{2}
71 1+(−6.74−3.89i)T+(35.5+61.4i)T2 1 + (-6.74 - 3.89i)T + (35.5 + 61.4i)T^{2}
73 1+4.30iT−73T2 1 + 4.30iT - 73T^{2}
79 1−13.4T+79T2 1 - 13.4T + 79T^{2}
83 1−17.6T+83T2 1 - 17.6T + 83T^{2}
89 1+(3.71−2.14i)T+(44.5−77.0i)T2 1 + (3.71 - 2.14i)T + (44.5 - 77.0i)T^{2}
97 1+(4.53+2.61i)T+(48.5+84.0i)T2 1 + (4.53 + 2.61i)T + (48.5 + 84.0i)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

−10.73648082825423416783643421447, −9.499530262607997576727409947995, −9.190371333166727499430666269064, −7.58350925118248985119230665949, −6.73557253543694378350327536474, −5.47614549781422740246415183479, −4.73384599514280157587526083629, −3.94794527682078830460641266474, −2.87313756505631227165150667034, −0.996533975007250013832133228443, 0.75371344320994192464180976210, 3.48136353320549791070632837452, 3.79947716929888785185269443363, 5.46565026989535750450445851466, 6.05367815905199268342621579702, 6.62882058390445998477612007350, 7.927026571353798390885645364917, 8.267619581235223800324750961037, 9.340757849669520238298144282056, 10.71809018853225300408064578181

Graph of the ZZ-function along the critical line