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Properties

Label 2-560-1.1-c1-0-2
Degree 22
Conductor 560560
Sign 11
Analytic cond. 4.471624.47162
Root an. cond. 2.114622.11462
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s − 2·9-s + 3·11-s + 5·13-s + 15-s + 3·17-s − 2·19-s + 21-s + 6·23-s + 25-s + 5·27-s + 3·29-s + 4·31-s − 3·33-s + 35-s + 2·37-s − 5·39-s − 12·41-s + 10·43-s + 2·45-s − 9·47-s + 49-s − 3·51-s + 12·53-s − 3·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.904·11-s + 1.38·13-s + 0.258·15-s + 0.727·17-s − 0.458·19-s + 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s + 0.718·31-s − 0.522·33-s + 0.169·35-s + 0.328·37-s − 0.800·39-s − 1.87·41-s + 1.52·43-s + 0.298·45-s − 1.31·47-s + 1/7·49-s − 0.420·51-s + 1.64·53-s − 0.404·55-s + ⋯

Functional equation

Λ(s)=(560s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(560s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 560560    =    24⋅5⋅72^{4} \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 4.471624.47162
Root analytic conductor: 2.114622.11462
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 560, ( :1/2), 1)(2,\ 560,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1025221381.102522138
L(12)L(\frac12) ≈\approx 1.1025221381.102522138
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 1
5 1+T 1 + T
7 1+T 1 + T
good3 1+T+pT2 1 + T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+12T+pT2 1 + 12 T + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1−T+pT2 1 - T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1+T+pT2 1 + T + p 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

−10.96922517526030591560500511793, −10.00137734072722000076745491348, −8.829468246037740149648448974500, −8.322081181544343689933636986956, −6.91010810779795429087322035266, −6.26051630300961999746999469622, −5.29742546009269090365766846421, −4.02723544557718882488682513095, −3.04856603410542935889592969685, −1.00806560019316329290903940266, 1.00806560019316329290903940266, 3.04856603410542935889592969685, 4.02723544557718882488682513095, 5.29742546009269090365766846421, 6.26051630300961999746999469622, 6.91010810779795429087322035266, 8.322081181544343689933636986956, 8.829468246037740149648448974500, 10.00137734072722000076745491348, 10.96922517526030591560500511793

Graph of the ZZ-function along the critical line