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Properties

Label 2-208-1.1-c9-0-31
Degree 22
Conductor 208208
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 47.8·3-s − 109.·5-s − 5.94e3·7-s − 1.73e4·9-s + 2.52e4·11-s + 2.85e4·13-s + 5.25e3·15-s + 1.09e5·17-s + 9.04e5·19-s + 2.84e5·21-s + 4.35e5·23-s − 1.94e6·25-s + 1.77e6·27-s + 6.44e6·29-s − 6.62e6·31-s − 1.20e6·33-s + 6.52e5·35-s + 4.14e6·37-s − 1.36e6·39-s + 1.49e7·41-s − 4.01e7·43-s + 1.90e6·45-s − 6.30e6·47-s − 4.98e6·49-s − 5.23e6·51-s + 1.53e7·53-s − 2.76e6·55-s + ⋯
L(s)  = 1  − 0.341·3-s − 0.0785·5-s − 0.936·7-s − 0.883·9-s + 0.519·11-s + 0.277·13-s + 0.0268·15-s + 0.317·17-s + 1.59·19-s + 0.319·21-s + 0.324·23-s − 0.993·25-s + 0.642·27-s + 1.69·29-s − 1.28·31-s − 0.177·33-s + 0.0735·35-s + 0.363·37-s − 0.0946·39-s + 0.826·41-s − 1.79·43-s + 0.0693·45-s − 0.188·47-s − 0.123·49-s − 0.108·51-s + 0.266·53-s − 0.0407·55-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: −1-1
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: 11
Selberg data: (2, 208, ( :9/2), −1)(2,\ 208,\ (\ :9/2),\ -1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
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+47.8T+1.96e4T2 1 + 47.8T + 1.96e4T^{2}
5 1+109.T+1.95e6T2 1 + 109.T + 1.95e6T^{2}
7 1+5.94e3T+4.03e7T2 1 + 5.94e3T + 4.03e7T^{2}
11 1−2.52e4T+2.35e9T2 1 - 2.52e4T + 2.35e9T^{2}
17 1−1.09e5T+1.18e11T2 1 - 1.09e5T + 1.18e11T^{2}
19 1−9.04e5T+3.22e11T2 1 - 9.04e5T + 3.22e11T^{2}
23 1−4.35e5T+1.80e12T2 1 - 4.35e5T + 1.80e12T^{2}
29 1−6.44e6T+1.45e13T2 1 - 6.44e6T + 1.45e13T^{2}
31 1+6.62e6T+2.64e13T2 1 + 6.62e6T + 2.64e13T^{2}
37 1−4.14e6T+1.29e14T2 1 - 4.14e6T + 1.29e14T^{2}
41 1−1.49e7T+3.27e14T2 1 - 1.49e7T + 3.27e14T^{2}
43 1+4.01e7T+5.02e14T2 1 + 4.01e7T + 5.02e14T^{2}
47 1+6.30e6T+1.11e15T2 1 + 6.30e6T + 1.11e15T^{2}
53 1−1.53e7T+3.29e15T2 1 - 1.53e7T + 3.29e15T^{2}
59 1−1.52e8T+8.66e15T2 1 - 1.52e8T + 8.66e15T^{2}
61 1−8.66e7T+1.16e16T2 1 - 8.66e7T + 1.16e16T^{2}
67 1−1.01e8T+2.72e16T2 1 - 1.01e8T + 2.72e16T^{2}
71 1+4.13e8T+4.58e16T2 1 + 4.13e8T + 4.58e16T^{2}
73 1+3.14e8T+5.88e16T2 1 + 3.14e8T + 5.88e16T^{2}
79 1−2.00e8T+1.19e17T2 1 - 2.00e8T + 1.19e17T^{2}
83 1+6.34e7T+1.86e17T2 1 + 6.34e7T + 1.86e17T^{2}
89 1−3.47e7T+3.50e17T2 1 - 3.47e7T + 3.50e17T^{2}
97 1+1.25e9T+7.60e17T2 1 + 1.25e9T + 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

−10.17141166420217528870790884133, −9.365241707898519012143571410896, −8.326281808654221457938322123238, −7.08077124576491817852382779598, −6.11263202233780018088132064070, −5.23371043423504310761354920702, −3.70088849799796462963140336257, −2.82386876762059450946972992543, −1.13933177173345191038160784256, 0, 1.13933177173345191038160784256, 2.82386876762059450946972992543, 3.70088849799796462963140336257, 5.23371043423504310761354920702, 6.11263202233780018088132064070, 7.08077124576491817852382779598, 8.326281808654221457938322123238, 9.365241707898519012143571410896, 10.17141166420217528870790884133

Graph of the ZZ-function along the critical line