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Properties

Label 2-208-1.1-c9-0-27
Degree 22
Conductor 208208
Sign 11
Analytic cond. 107.127107.127
Root an. cond. 10.350210.3502
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 204.·3-s − 258.·5-s + 8.86e3·7-s + 2.21e4·9-s − 3.60e4·11-s − 2.85e4·13-s − 5.29e4·15-s + 3.27e5·17-s − 2.65e5·19-s + 1.81e6·21-s + 2.42e6·23-s − 1.88e6·25-s + 5.09e5·27-s + 3.99e6·29-s + 6.45e6·31-s − 7.38e6·33-s − 2.29e6·35-s − 8.15e6·37-s − 5.84e6·39-s − 7.20e5·41-s + 4.13e7·43-s − 5.74e6·45-s − 3.67e7·47-s + 3.81e7·49-s + 6.69e7·51-s + 1.67e7·53-s + 9.34e6·55-s + ⋯
L(s)  = 1  + 1.45·3-s − 0.185·5-s + 1.39·7-s + 1.12·9-s − 0.743·11-s − 0.277·13-s − 0.270·15-s + 0.950·17-s − 0.467·19-s + 2.03·21-s + 1.80·23-s − 0.965·25-s + 0.184·27-s + 1.04·29-s + 1.25·31-s − 1.08·33-s − 0.258·35-s − 0.715·37-s − 0.404·39-s − 0.0398·41-s + 1.84·43-s − 0.208·45-s − 1.09·47-s + 0.946·49-s + 1.38·51-s + 0.291·53-s + 0.137·55-s + ⋯

Functional equation

Λ(s)=(208s/2 ΓC(s) L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(208s/2 ΓC(s+9/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 208208    =    24⋅132^{4} \cdot 13
Sign: 11
Analytic conductor: 107.127107.127
Root analytic conductor: 10.350210.3502
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 208, ( :9/2), 1)(2,\ 208,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 4.6591873764.659187376
L(12)L(\frac12) ≈\approx 4.6591873764.659187376
L(112)L(\frac{11}{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
13 1+2.85e4T 1 + 2.85e4T
good3 1−204.T+1.96e4T2 1 - 204.T + 1.96e4T^{2}
5 1+258.T+1.95e6T2 1 + 258.T + 1.95e6T^{2}
7 1−8.86e3T+4.03e7T2 1 - 8.86e3T + 4.03e7T^{2}
11 1+3.60e4T+2.35e9T2 1 + 3.60e4T + 2.35e9T^{2}
17 1−3.27e5T+1.18e11T2 1 - 3.27e5T + 1.18e11T^{2}
19 1+2.65e5T+3.22e11T2 1 + 2.65e5T + 3.22e11T^{2}
23 1−2.42e6T+1.80e12T2 1 - 2.42e6T + 1.80e12T^{2}
29 1−3.99e6T+1.45e13T2 1 - 3.99e6T + 1.45e13T^{2}
31 1−6.45e6T+2.64e13T2 1 - 6.45e6T + 2.64e13T^{2}
37 1+8.15e6T+1.29e14T2 1 + 8.15e6T + 1.29e14T^{2}
41 1+7.20e5T+3.27e14T2 1 + 7.20e5T + 3.27e14T^{2}
43 1−4.13e7T+5.02e14T2 1 - 4.13e7T + 5.02e14T^{2}
47 1+3.67e7T+1.11e15T2 1 + 3.67e7T + 1.11e15T^{2}
53 1−1.67e7T+3.29e15T2 1 - 1.67e7T + 3.29e15T^{2}
59 1−5.89e7T+8.66e15T2 1 - 5.89e7T + 8.66e15T^{2}
61 1−1.28e8T+1.16e16T2 1 - 1.28e8T + 1.16e16T^{2}
67 1−1.91e8T+2.72e16T2 1 - 1.91e8T + 2.72e16T^{2}
71 1+3.40e8T+4.58e16T2 1 + 3.40e8T + 4.58e16T^{2}
73 1−3.19e8T+5.88e16T2 1 - 3.19e8T + 5.88e16T^{2}
79 1+2.44e8T+1.19e17T2 1 + 2.44e8T + 1.19e17T^{2}
83 1−4.38e8T+1.86e17T2 1 - 4.38e8T + 1.86e17T^{2}
89 1+7.90e8T+3.50e17T2 1 + 7.90e8T + 3.50e17T^{2}
97 1+1.65e8T+7.60e17T2 1 + 1.65e8T + 7.60e17T^{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

−10.64334664654949697392773588705, −9.619079419077083341211188302115, −8.434423998729304566430308474066, −8.070923609894968178564335892853, −7.13966175791632708156159976087, −5.33578797336076853350160111911, −4.34635727119769224519790553274, −3.06655046904912396420465479271, −2.19274826127070095583355114948, −1.01077646326770093102294153631, 1.01077646326770093102294153631, 2.19274826127070095583355114948, 3.06655046904912396420465479271, 4.34635727119769224519790553274, 5.33578797336076853350160111911, 7.13966175791632708156159976087, 8.070923609894968178564335892853, 8.434423998729304566430308474066, 9.619079419077083341211188302115, 10.64334664654949697392773588705

Graph of the ZZ-function along the critical line