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Properties

Label 2-208-1.1-c9-0-38
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  − 250.·3-s + 1.55e3·5-s + 8.32e3·7-s + 4.30e4·9-s − 3.07e4·11-s + 2.85e4·13-s − 3.89e5·15-s − 6.37e5·17-s − 1.05e5·19-s − 2.08e6·21-s + 5.11e5·23-s + 4.66e5·25-s − 5.85e6·27-s + 7.81e5·29-s + 2.83e6·31-s + 7.70e6·33-s + 1.29e7·35-s + 1.22e7·37-s − 7.15e6·39-s − 6.83e6·41-s − 3.84e7·43-s + 6.69e7·45-s + 1.30e7·47-s + 2.90e7·49-s + 1.59e8·51-s − 2.42e7·53-s − 4.78e7·55-s + ⋯
L(s)  = 1  − 1.78·3-s + 1.11·5-s + 1.31·7-s + 2.18·9-s − 0.633·11-s + 0.277·13-s − 1.98·15-s − 1.85·17-s − 0.186·19-s − 2.34·21-s + 0.380·23-s + 0.238·25-s − 2.12·27-s + 0.205·29-s + 0.550·31-s + 1.13·33-s + 1.45·35-s + 1.07·37-s − 0.495·39-s − 0.377·41-s − 1.71·43-s + 2.43·45-s + 0.389·47-s + 0.719·49-s + 3.30·51-s − 0.422·53-s − 0.704·55-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+250.T+1.96e4T2 1 + 250.T + 1.96e4T^{2}
5 1−1.55e3T+1.95e6T2 1 - 1.55e3T + 1.95e6T^{2}
7 1−8.32e3T+4.03e7T2 1 - 8.32e3T + 4.03e7T^{2}
11 1+3.07e4T+2.35e9T2 1 + 3.07e4T + 2.35e9T^{2}
17 1+6.37e5T+1.18e11T2 1 + 6.37e5T + 1.18e11T^{2}
19 1+1.05e5T+3.22e11T2 1 + 1.05e5T + 3.22e11T^{2}
23 1−5.11e5T+1.80e12T2 1 - 5.11e5T + 1.80e12T^{2}
29 1−7.81e5T+1.45e13T2 1 - 7.81e5T + 1.45e13T^{2}
31 1−2.83e6T+2.64e13T2 1 - 2.83e6T + 2.64e13T^{2}
37 1−1.22e7T+1.29e14T2 1 - 1.22e7T + 1.29e14T^{2}
41 1+6.83e6T+3.27e14T2 1 + 6.83e6T + 3.27e14T^{2}
43 1+3.84e7T+5.02e14T2 1 + 3.84e7T + 5.02e14T^{2}
47 1−1.30e7T+1.11e15T2 1 - 1.30e7T + 1.11e15T^{2}
53 1+2.42e7T+3.29e15T2 1 + 2.42e7T + 3.29e15T^{2}
59 1+1.63e8T+8.66e15T2 1 + 1.63e8T + 8.66e15T^{2}
61 1−1.90e7T+1.16e16T2 1 - 1.90e7T + 1.16e16T^{2}
67 1−7.22e7T+2.72e16T2 1 - 7.22e7T + 2.72e16T^{2}
71 1+2.65e7T+4.58e16T2 1 + 2.65e7T + 4.58e16T^{2}
73 1−2.42e8T+5.88e16T2 1 - 2.42e8T + 5.88e16T^{2}
79 1−4.64e8T+1.19e17T2 1 - 4.64e8T + 1.19e17T^{2}
83 1+5.46e8T+1.86e17T2 1 + 5.46e8T + 1.86e17T^{2}
89 1−3.65e8T+3.50e17T2 1 - 3.65e8T + 3.50e17T^{2}
97 1−9.98e7T+7.60e17T2 1 - 9.98e7T + 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

−10.66704370040835682598586082630, −9.551777206714869085647467455149, −8.201451613993912249269769036199, −6.80910038270533237670810923221, −6.05247961861548853021674216887, −5.08359343174844389881588039583, −4.53054225710761360075774885240, −2.16573680279845600868060409414, −1.26658690714945075218574020483, 0, 1.26658690714945075218574020483, 2.16573680279845600868060409414, 4.53054225710761360075774885240, 5.08359343174844389881588039583, 6.05247961861548853021674216887, 6.80910038270533237670810923221, 8.201451613993912249269769036199, 9.551777206714869085647467455149, 10.66704370040835682598586082630

Graph of the ZZ-function along the critical line