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Properties

Label 2-360-5.4-c5-0-12
Degree 22
Conductor 360360
Sign 0.795−0.605i0.795 - 0.605i
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  + (−33.8 − 44.4i)5-s + 85.8i·7-s + 488.·11-s − 38.5i·13-s + 692. i·17-s − 2.48e3·19-s − 4.12e3i·23-s + (−830. + 3.01e3i)25-s − 1.88e3·29-s − 2.29e3·31-s + (3.81e3 − 2.90e3i)35-s + 1.06e4i·37-s + 1.52e4·41-s + 9.46e3i·43-s + 1.43e4i·47-s + ⋯
L(s)  = 1  + (−0.605 − 0.795i)5-s + 0.662i·7-s + 1.21·11-s − 0.0633i·13-s + 0.581i·17-s − 1.58·19-s − 1.62i·23-s + (−0.265 + 0.964i)25-s − 0.416·29-s − 0.429·31-s + (0.526 − 0.401i)35-s + 1.27i·37-s + 1.41·41-s + 0.780i·43-s + 0.945i·47-s + ⋯

Functional equation

Λ(s)=(360s/2 ΓC(s) L(s)=((0.795−0.605i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(360s/2 ΓC(s+5/2) L(s)=((0.795−0.605i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 360360    =    23⋅32⋅52^{3} \cdot 3^{2} \cdot 5
Sign: 0.795−0.605i0.795 - 0.605i
Analytic conductor: 57.738157.7381
Root analytic conductor: 7.598567.59856
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ360(289,⋅)\chi_{360} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 360, ( :5/2), 0.795−0.605i)(2,\ 360,\ (\ :5/2),\ 0.795 - 0.605i)

Particular Values

L(3)L(3) ≈\approx 1.5823074701.582307470
L(12)L(\frac12) ≈\approx 1.5823074701.582307470
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 1
3 1 1
5 1+(33.8+44.4i)T 1 + (33.8 + 44.4i)T
good7 1−85.8iT−1.68e4T2 1 - 85.8iT - 1.68e4T^{2}
11 1−488.T+1.61e5T2 1 - 488.T + 1.61e5T^{2}
13 1+38.5iT−3.71e5T2 1 + 38.5iT - 3.71e5T^{2}
17 1−692.iT−1.41e6T2 1 - 692. iT - 1.41e6T^{2}
19 1+2.48e3T+2.47e6T2 1 + 2.48e3T + 2.47e6T^{2}
23 1+4.12e3iT−6.43e6T2 1 + 4.12e3iT - 6.43e6T^{2}
29 1+1.88e3T+2.05e7T2 1 + 1.88e3T + 2.05e7T^{2}
31 1+2.29e3T+2.86e7T2 1 + 2.29e3T + 2.86e7T^{2}
37 1−1.06e4iT−6.93e7T2 1 - 1.06e4iT - 6.93e7T^{2}
41 1−1.52e4T+1.15e8T2 1 - 1.52e4T + 1.15e8T^{2}
43 1−9.46e3iT−1.47e8T2 1 - 9.46e3iT - 1.47e8T^{2}
47 1−1.43e4iT−2.29e8T2 1 - 1.43e4iT - 2.29e8T^{2}
53 1+1.39e4iT−4.18e8T2 1 + 1.39e4iT - 4.18e8T^{2}
59 1−3.96e4T+7.14e8T2 1 - 3.96e4T + 7.14e8T^{2}
61 1−3.82e4T+8.44e8T2 1 - 3.82e4T + 8.44e8T^{2}
67 1+1.41e3iT−1.35e9T2 1 + 1.41e3iT - 1.35e9T^{2}
71 1−1.16e4T+1.80e9T2 1 - 1.16e4T + 1.80e9T^{2}
73 1−3.10e4iT−2.07e9T2 1 - 3.10e4iT - 2.07e9T^{2}
79 1−4.34e4T+3.07e9T2 1 - 4.34e4T + 3.07e9T^{2}
83 1−8.66e4iT−3.93e9T2 1 - 8.66e4iT - 3.93e9T^{2}
89 1+1.00e5T+5.58e9T2 1 + 1.00e5T + 5.58e9T^{2}
97 1+3.99e4iT−8.58e9T2 1 + 3.99e4iT - 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

−10.86800755948972199153788655248, −9.615402367229056251377435755519, −8.675928687836215873975190721244, −8.258302016096967383287421567486, −6.80238629942353560790000864776, −5.92997678749114484296412637477, −4.60855020606864027673625494701, −3.86166044105265330315524953504, −2.24363325202322463019501312745, −0.898356521068813594849341273214, 0.52234317014571648782525174908, 2.08392289356223263064983951974, 3.62753032042547365756787678069, 4.18209174092434816452245508904, 5.80388977313753354510710518870, 6.93305739044423728075365321679, 7.42986202262722750502223504425, 8.684797896448448906781313543168, 9.632864933024185725898264264655, 10.70212640420696129953356005635

Graph of the ZZ-function along the critical line