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Properties

Label 2-4400-1.1-c1-0-66
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  − 2·7-s − 3·9-s − 11-s + 4·13-s + 4·17-s − 6·29-s + 2·37-s + 6·41-s + 2·43-s − 3·49-s + 10·53-s − 12·59-s − 6·61-s + 6·63-s − 12·67-s − 16·71-s − 4·73-s + 2·77-s + 4·79-s + 9·81-s + 2·83-s + 6·89-s − 8·91-s + 2·97-s + 3·99-s + 6·101-s + 4·103-s + ⋯
L(s)  = 1  − 0.755·7-s − 9-s − 0.301·11-s + 1.10·13-s + 0.970·17-s − 1.11·29-s + 0.328·37-s + 0.937·41-s + 0.304·43-s − 3/7·49-s + 1.37·53-s − 1.56·59-s − 0.768·61-s + 0.755·63-s − 1.46·67-s − 1.89·71-s − 0.468·73-s + 0.227·77-s + 0.450·79-s + 81-s + 0.219·83-s + 0.635·89-s − 0.838·91-s + 0.203·97-s + 0.301·99-s + 0.597·101-s + 0.394·103-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+pT2 1 + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1+16T+pT2 1 + 16 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−2T+pT2 1 - 2 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−7.898936570941772463430241678285, −7.44620724213555121019017614969, −6.23862547700277970583253282336, −5.97420845383247042083925636614, −5.19222357781999791105637954091, −4.04649274617805365151136230133, −3.30588833879895129178850964400, −2.64602945319110350841739354126, −1.31982113773072061232939185969, 0, 1.31982113773072061232939185969, 2.64602945319110350841739354126, 3.30588833879895129178850964400, 4.04649274617805365151136230133, 5.19222357781999791105637954091, 5.97420845383247042083925636614, 6.23862547700277970583253282336, 7.44620724213555121019017614969, 7.898936570941772463430241678285

Graph of the ZZ-function along the critical line