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Properties

Label 2-4400-1.1-c1-0-93
Degree 22
Conductor 44004400
Sign 11
Analytic cond. 35.134135.1341
Root an. cond. 5.927405.92740
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·7-s + 9-s − 11-s − 6·13-s + 2·17-s − 4·19-s + 8·21-s − 6·23-s + 4·27-s − 2·29-s − 8·31-s + 2·33-s − 8·37-s + 12·39-s + 6·41-s − 12·43-s − 10·47-s + 9·49-s − 4·51-s + 8·57-s + 4·59-s − 10·61-s − 4·63-s − 2·67-s + 12·69-s + 8·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.485·17-s − 0.917·19-s + 1.74·21-s − 1.25·23-s + 0.769·27-s − 0.371·29-s − 1.43·31-s + 0.348·33-s − 1.31·37-s + 1.92·39-s + 0.937·41-s − 1.82·43-s − 1.45·47-s + 9/7·49-s − 0.560·51-s + 1.05·57-s + 0.520·59-s − 1.28·61-s − 0.503·63-s − 0.244·67-s + 1.44·69-s + 0.949·71-s + ⋯

Functional equation

Λ(s)=(4400s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4400s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 44004400    =    24⋅52⋅112^{4} \cdot 5^{2} \cdot 11
Sign: 11
Analytic conductor: 35.134135.1341
Root analytic conductor: 5.927405.92740
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 22
Selberg data: (2, 4400, ( :1/2), 1)(2,\ 4400,\ (\ :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
5 1 1
11 1+T 1 + T
good3 1+2T+pT2 1 + 2 T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+12T+pT2 1 + 12 T + p T^{2}
47 1+10T+pT2 1 + 10 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1+4T+pT2 1 + 4 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

−7.39670640039587408191046820043, −6.76524121744274112910508903739, −6.17312501935864596969907723956, −5.47088967195342583565929028119, −4.85974732655216967430322002334, −3.80606683850703192972973716479, −2.96142857643921751826612429350, −1.93482862156308380196244786119, 0, 0, 1.93482862156308380196244786119, 2.96142857643921751826612429350, 3.80606683850703192972973716479, 4.85974732655216967430322002334, 5.47088967195342583565929028119, 6.17312501935864596969907723956, 6.76524121744274112910508903739, 7.39670640039587408191046820043

Graph of the ZZ-function along the critical line