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Properties

Label 2-360-8.5-c5-0-22
Degree 22
Conductor 360360
Sign −0.993−0.116i-0.993 - 0.116i
Analytic cond. 57.738157.7381
Root an. cond. 7.598567.59856
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.78 + 3.01i)2-s + (13.7 + 28.8i)4-s − 25i·5-s − 56.4·7-s + (−21.0 + 179. i)8-s + (75.4 − 119. i)10-s + 261. i·11-s + 720. i·13-s + (−270. − 170. i)14-s + (−643. + 796. i)16-s + 1.87e3·17-s − 1.99e3i·19-s + (721. − 344. i)20-s + (−787. + 1.24e3i)22-s − 2.57e3·23-s + ⋯
L(s)  = 1  + (0.845 + 0.533i)2-s + (0.431 + 0.902i)4-s − 0.447i·5-s − 0.435·7-s + (−0.116 + 0.993i)8-s + (0.238 − 0.378i)10-s + 0.650i·11-s + 1.18i·13-s + (−0.368 − 0.232i)14-s + (−0.628 + 0.778i)16-s + 1.57·17-s − 1.26i·19-s + (0.403 − 0.192i)20-s + (−0.346 + 0.550i)22-s − 1.01·23-s + ⋯

Functional equation

Λ(s)=(360s/2 ΓC(s) L(s)=((−0.993−0.116i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(360s/2 ΓC(s+5/2) L(s)=((−0.993−0.116i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 360360    =    23⋅32⋅52^{3} \cdot 3^{2} \cdot 5
Sign: −0.993−0.116i-0.993 - 0.116i
Analytic conductor: 57.738157.7381
Root analytic conductor: 7.598567.59856
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ360(181,⋅)\chi_{360} (181, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 360, ( :5/2), −0.993−0.116i)(2,\ 360,\ (\ :5/2),\ -0.993 - 0.116i)

Particular Values

L(3)L(3) ≈\approx 1.8937893901.893789390
L(12)L(\frac12) ≈\approx 1.8937893901.893789390
L(72)L(\frac{7}{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+(−4.78−3.01i)T 1 + (-4.78 - 3.01i)T
3 1 1
5 1+25iT 1 + 25iT
good7 1+56.4T+1.68e4T2 1 + 56.4T + 1.68e4T^{2}
11 1−261.iT−1.61e5T2 1 - 261. iT - 1.61e5T^{2}
13 1−720.iT−3.71e5T2 1 - 720. iT - 3.71e5T^{2}
17 1−1.87e3T+1.41e6T2 1 - 1.87e3T + 1.41e6T^{2}
19 1+1.99e3iT−2.47e6T2 1 + 1.99e3iT - 2.47e6T^{2}
23 1+2.57e3T+6.43e6T2 1 + 2.57e3T + 6.43e6T^{2}
29 1+1.70e3iT−2.05e7T2 1 + 1.70e3iT - 2.05e7T^{2}
31 1+7.73e3T+2.86e7T2 1 + 7.73e3T + 2.86e7T^{2}
37 1−1.22e4iT−6.93e7T2 1 - 1.22e4iT - 6.93e7T^{2}
41 1+1.49e4T+1.15e8T2 1 + 1.49e4T + 1.15e8T^{2}
43 1−1.81e4iT−1.47e8T2 1 - 1.81e4iT - 1.47e8T^{2}
47 1+2.14e3T+2.29e8T2 1 + 2.14e3T + 2.29e8T^{2}
53 1+1.60e3iT−4.18e8T2 1 + 1.60e3iT - 4.18e8T^{2}
59 1+2.68e3iT−7.14e8T2 1 + 2.68e3iT - 7.14e8T^{2}
61 1−4.45e4iT−8.44e8T2 1 - 4.45e4iT - 8.44e8T^{2}
67 1+1.24e4iT−1.35e9T2 1 + 1.24e4iT - 1.35e9T^{2}
71 1+8.18e3T+1.80e9T2 1 + 8.18e3T + 1.80e9T^{2}
73 1+4.10e4T+2.07e9T2 1 + 4.10e4T + 2.07e9T^{2}
79 1−4.63e4T+3.07e9T2 1 - 4.63e4T + 3.07e9T^{2}
83 1+6.16e4iT−3.93e9T2 1 + 6.16e4iT - 3.93e9T^{2}
89 1+5.32e4T+5.58e9T2 1 + 5.32e4T + 5.58e9T^{2}
97 1+3.92e4T+8.58e9T2 1 + 3.92e4T + 8.58e9T^{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

−11.49278957330199262281645114410, −10.02888605630489800997346391492, −9.147102279131487418021280779502, −8.041244713476131031367194288282, −7.11495678030438707807720204463, −6.24645887524989864403480775960, −5.13219339965880535620367483661, −4.26699966708305401168219599983, −3.12421158080268826447540289508, −1.71334156750915753822295096510, 0.32724985338002001614988813868, 1.77584305025505061400310945511, 3.25650353783891998038891973743, 3.67652534147793455630545132513, 5.49150911436203420454280454933, 5.85121668201411534462030908350, 7.17183890756217614105776189799, 8.195665420382237564243755648096, 9.707208163288533357170667149913, 10.31729815656893805297178151507

Graph of the ZZ-function along the critical line