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Properties

Label 2-360-5.4-c3-0-8
Degree 22
Conductor 360360
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 21.240621.2406
Root an. cond. 4.608764.60876
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10 + 5i)5-s + 10i·7-s + 46·11-s + 34i·13-s − 66i·17-s − 104·19-s + 164i·23-s + (75 + 100i)25-s + 224·29-s − 72·31-s + (−50 + 100i)35-s − 22i·37-s − 194·41-s − 108i·43-s + 480i·47-s + ⋯
L(s)  = 1  + (0.894 + 0.447i)5-s + 0.539i·7-s + 1.26·11-s + 0.725i·13-s − 0.941i·17-s − 1.25·19-s + 1.48i·23-s + (0.599 + 0.800i)25-s + 1.43·29-s − 0.417·31-s + (−0.241 + 0.482i)35-s − 0.0977i·37-s − 0.738·41-s − 0.383i·43-s + 1.48i·47-s + ⋯

Functional equation

Λ(s)=(360s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(360s/2 ΓC(s+3/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 360360    =    23⋅32⋅52^{3} \cdot 3^{2} \cdot 5
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 21.240621.2406
Root analytic conductor: 4.608764.60876
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ360(289,⋅)\chi_{360} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 360, ( :3/2), 0.447−0.894i)(2,\ 360,\ (\ :3/2),\ 0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 2.2320797662.232079766
L(12)L(\frac12) ≈\approx 2.2320797662.232079766
L(52)L(\frac{5}{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 1
3 1 1
5 1+(−10−5i)T 1 + (-10 - 5i)T
good7 1−10iT−343T2 1 - 10iT - 343T^{2}
11 1−46T+1.33e3T2 1 - 46T + 1.33e3T^{2}
13 1−34iT−2.19e3T2 1 - 34iT - 2.19e3T^{2}
17 1+66iT−4.91e3T2 1 + 66iT - 4.91e3T^{2}
19 1+104T+6.85e3T2 1 + 104T + 6.85e3T^{2}
23 1−164iT−1.21e4T2 1 - 164iT - 1.21e4T^{2}
29 1−224T+2.43e4T2 1 - 224T + 2.43e4T^{2}
31 1+72T+2.97e4T2 1 + 72T + 2.97e4T^{2}
37 1+22iT−5.06e4T2 1 + 22iT - 5.06e4T^{2}
41 1+194T+6.89e4T2 1 + 194T + 6.89e4T^{2}
43 1+108iT−7.95e4T2 1 + 108iT - 7.95e4T^{2}
47 1−480iT−1.03e5T2 1 - 480iT - 1.03e5T^{2}
53 1−286iT−1.48e5T2 1 - 286iT - 1.48e5T^{2}
59 1−426T+2.05e5T2 1 - 426T + 2.05e5T^{2}
61 1−698T+2.26e5T2 1 - 698T + 2.26e5T^{2}
67 1−328iT−3.00e5T2 1 - 328iT - 3.00e5T^{2}
71 1+188T+3.57e5T2 1 + 188T + 3.57e5T^{2}
73 1−740iT−3.89e5T2 1 - 740iT - 3.89e5T^{2}
79 1+1.16e3T+4.93e5T2 1 + 1.16e3T + 4.93e5T^{2}
83 1−412iT−5.71e5T2 1 - 412iT - 5.71e5T^{2}
89 1−1.20e3T+7.04e5T2 1 - 1.20e3T + 7.04e5T^{2}
97 1+1.38e3iT−9.12e5T2 1 + 1.38e3iT - 9.12e5T^{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

−11.29538145624414838549198783694, −10.12871929808344466932678234674, −9.313642687639383407151901736328, −8.684455406410368199689175610371, −7.11514844486315936926220172683, −6.42409931247853907755329678899, −5.44267021980031410157725107356, −4.11945471170727165535028091224, −2.68614745937563140591840769981, −1.49789329027816537129462274198, 0.834383004547964715258259548850, 2.15379328365932639722440774061, 3.81778435177005600299055533724, 4.85157262227652770006547910302, 6.17964246343205437727121460810, 6.74261670605340473633676975745, 8.345636407036975257591176369287, 8.852793287731828293621754148733, 10.19301282663833798436941161575, 10.49143788510375611610392382679

Graph of the ZZ-function along the critical line