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Properties

Label 2-4400-1.1-c1-0-65
Degree 22
Conductor 44004400
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s − 2·9-s + 11-s − 17-s − 19-s − 21-s + 5·27-s − 29-s + 31-s − 33-s − 37-s + 6·43-s + 8·47-s − 6·49-s + 51-s − 9·53-s + 57-s − 4·59-s − 7·61-s − 2·63-s − 4·67-s − 5·71-s − 14·73-s + 77-s − 4·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.301·11-s − 0.242·17-s − 0.229·19-s − 0.218·21-s + 0.962·27-s − 0.185·29-s + 0.179·31-s − 0.174·33-s − 0.164·37-s + 0.914·43-s + 1.16·47-s − 6/7·49-s + 0.140·51-s − 1.23·53-s + 0.132·57-s − 0.520·59-s − 0.896·61-s − 0.251·63-s − 0.488·67-s − 0.593·71-s − 1.63·73-s + 0.113·77-s − 0.450·79-s + 1/9·81-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: −1-1
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: 11
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+T+pT2 1 + T + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+T+pT2 1 + T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+T+pT2 1 + T + p T^{2}
31 1−T+pT2 1 - T + p T^{2}
37 1+T+pT2 1 + T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+9T+pT2 1 + 9 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+5T+pT2 1 + 5 T + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1+7T+pT2 1 + 7 T + p T^{2}
97 1−16T+pT2 1 - 16 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

−7.951662380660049215969666649110, −7.29997142715825538555598519161, −6.31897820911954060047884905099, −5.92559031307278184228698268950, −5.00662765949998344422587714046, −4.39602918659409123463354737339, −3.35707500162431374995492464731, −2.42985027556468358842264902413, −1.29861350723040713076403758694, 0, 1.29861350723040713076403758694, 2.42985027556468358842264902413, 3.35707500162431374995492464731, 4.39602918659409123463354737339, 5.00662765949998344422587714046, 5.92559031307278184228698268950, 6.31897820911954060047884905099, 7.29997142715825538555598519161, 7.951662380660049215969666649110

Graph of the ZZ-function along the critical line