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Properties

Label 2-448-1.1-c1-0-9
Degree 22
Conductor 448448
Sign −1-1
Analytic cond. 3.577293.57729
Root an. cond. 1.891371.89137
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 7-s − 3·9-s + 4·11-s − 2·13-s − 6·17-s − 8·19-s − 25-s − 6·29-s + 8·31-s + 2·35-s + 2·37-s + 2·41-s + 4·43-s + 6·45-s − 8·47-s + 49-s − 6·53-s − 8·55-s + 6·61-s + 3·63-s + 4·65-s + 4·67-s − 8·71-s + 10·73-s − 4·77-s + 16·79-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 9-s + 1.20·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.338·35-s + 0.328·37-s + 0.312·41-s + 0.609·43-s + 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s + 0.768·61-s + 0.377·63-s + 0.496·65-s + 0.488·67-s − 0.949·71-s + 1.17·73-s − 0.455·77-s + 1.80·79-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: −1-1
Analytic conductor: 3.577293.57729
Root analytic conductor: 1.891371.89137
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 448, ( :1/2), −1)(2,\ 448,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
7 1+T 1 + T
good3 1+pT2 1 + p T^{2}
5 1+2T+pT2 1 + 2 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1+8T+pT2 1 + 8 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+6T+pT2 1 + 6 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

−10.95242742749570979564901530187, −9.609035860411891137058317154686, −8.773870329302691374695991117300, −8.056242270008326333596348707107, −6.78746217059840482908609994194, −6.15054397459112152814277916997, −4.58367495132727840614913978641, −3.77515777589665000060106383714, −2.35911554344998475560129433401, 0, 2.35911554344998475560129433401, 3.77515777589665000060106383714, 4.58367495132727840614913978641, 6.15054397459112152814277916997, 6.78746217059840482908609994194, 8.056242270008326333596348707107, 8.773870329302691374695991117300, 9.609035860411891137058317154686, 10.95242742749570979564901530187

Graph of the ZZ-function along the critical line