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Properties

Label 2-84-1.1-c5-0-3
Degree 22
Conductor 8484
Sign 11
Analytic cond. 13.472213.4722
Root an. cond. 3.670453.67045
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s + 106.·5-s + 49·7-s + 81·9-s − 250.·11-s − 300.·13-s + 957.·15-s + 2.02e3·17-s − 2.25e3·19-s + 441·21-s + 3.09e3·23-s + 8.19e3·25-s + 729·27-s − 6.60e3·29-s + 833.·31-s − 2.25e3·33-s + 5.21e3·35-s + 8.95e3·37-s − 2.70e3·39-s − 7.20e3·41-s + 1.44e4·43-s + 8.61e3·45-s − 1.79e4·47-s + 2.40e3·49-s + 1.81e4·51-s − 1.58e4·53-s − 2.66e4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.90·5-s + 0.377·7-s + 0.333·9-s − 0.623·11-s − 0.493·13-s + 1.09·15-s + 1.69·17-s − 1.43·19-s + 0.218·21-s + 1.21·23-s + 2.62·25-s + 0.192·27-s − 1.45·29-s + 0.155·31-s − 0.359·33-s + 0.719·35-s + 1.07·37-s − 0.284·39-s − 0.669·41-s + 1.19·43-s + 0.634·45-s − 1.18·47-s + 0.142·49-s + 0.979·51-s − 0.773·53-s − 1.18·55-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: 11
Analytic conductor: 13.472213.4722
Root analytic conductor: 3.670453.67045
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 84, ( :5/2), 1)(2,\ 84,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.9909110792.990911079
L(12)L(\frac12) ≈\approx 2.9909110792.990911079
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 1−9T 1 - 9T
7 1−49T 1 - 49T
good5 1−106.T+3.12e3T2 1 - 106.T + 3.12e3T^{2}
11 1+250.T+1.61e5T2 1 + 250.T + 1.61e5T^{2}
13 1+300.T+3.71e5T2 1 + 300.T + 3.71e5T^{2}
17 1−2.02e3T+1.41e6T2 1 - 2.02e3T + 1.41e6T^{2}
19 1+2.25e3T+2.47e6T2 1 + 2.25e3T + 2.47e6T^{2}
23 1−3.09e3T+6.43e6T2 1 - 3.09e3T + 6.43e6T^{2}
29 1+6.60e3T+2.05e7T2 1 + 6.60e3T + 2.05e7T^{2}
31 1−833.T+2.86e7T2 1 - 833.T + 2.86e7T^{2}
37 1−8.95e3T+6.93e7T2 1 - 8.95e3T + 6.93e7T^{2}
41 1+7.20e3T+1.15e8T2 1 + 7.20e3T + 1.15e8T^{2}
43 1−1.44e4T+1.47e8T2 1 - 1.44e4T + 1.47e8T^{2}
47 1+1.79e4T+2.29e8T2 1 + 1.79e4T + 2.29e8T^{2}
53 1+1.58e4T+4.18e8T2 1 + 1.58e4T + 4.18e8T^{2}
59 1+2.67e4T+7.14e8T2 1 + 2.67e4T + 7.14e8T^{2}
61 1+2.67e4T+8.44e8T2 1 + 2.67e4T + 8.44e8T^{2}
67 1+4.44e4T+1.35e9T2 1 + 4.44e4T + 1.35e9T^{2}
71 1+2.04e4T+1.80e9T2 1 + 2.04e4T + 1.80e9T^{2}
73 1−3.87e4T+2.07e9T2 1 - 3.87e4T + 2.07e9T^{2}
79 1+6.72e4T+3.07e9T2 1 + 6.72e4T + 3.07e9T^{2}
83 1+3.58e4T+3.93e9T2 1 + 3.58e4T + 3.93e9T^{2}
89 1−1.06e5T+5.58e9T2 1 - 1.06e5T + 5.58e9T^{2}
97 1−9.81e4T+8.58e9T2 1 - 9.81e4T + 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

−13.30371829236956142135149667851, −12.66447960539359114067315176878, −10.77972170603551267427942591296, −9.889446047578646413826003908938, −9.016983762345606754708982675844, −7.62200588550972514227763524971, −6.09872618699120467039700897528, −4.99941570222576614436449275870, −2.79822937856559072442082209064, −1.58768645208602917670190450878, 1.58768645208602917670190450878, 2.79822937856559072442082209064, 4.99941570222576614436449275870, 6.09872618699120467039700897528, 7.62200588550972514227763524971, 9.016983762345606754708982675844, 9.889446047578646413826003908938, 10.77972170603551267427942591296, 12.66447960539359114067315176878, 13.30371829236956142135149667851

Graph of the ZZ-function along the critical line