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Properties

Label 2-880-1.1-c1-0-19
Degree 22
Conductor 880880
Sign −1-1
Analytic cond. 7.026837.02683
Root an. cond. 2.650812.65081
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 + 5-s − 3·7-s − 2·9-s − 11-s − 6·13-s + 15-s − 7·17-s − 5·19-s − 3·21-s + 6·23-s + 25-s − 5·27-s + 5·29-s + 3·31-s − 33-s − 3·35-s + 3·37-s − 6·39-s + 2·41-s − 4·43-s − 2·45-s + 2·47-s + 2·49-s − 7·51-s − 53-s − 55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.13·7-s − 2/3·9-s − 0.301·11-s − 1.66·13-s + 0.258·15-s − 1.69·17-s − 1.14·19-s − 0.654·21-s + 1.25·23-s + 1/5·25-s − 0.962·27-s + 0.928·29-s + 0.538·31-s − 0.174·33-s − 0.507·35-s + 0.493·37-s − 0.960·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s + 0.291·47-s + 2/7·49-s − 0.980·51-s − 0.137·53-s − 0.134·55-s + ⋯

Functional equation

Λ(s)=(880s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(880s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 880880    =    24⋅5⋅112^{4} \cdot 5 \cdot 11
Sign: −1-1
Analytic conductor: 7.026837.02683
Root analytic conductor: 2.650812.65081
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 880, ( :1/2), −1)(2,\ 880,\ (\ :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−T 1 - T
11 1+T 1 + T
good3 1−T+pT2 1 - T + p T^{2}
7 1+3T+pT2 1 + 3 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1+7T+pT2 1 + 7 T + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−5T+pT2 1 - 5 T + p T^{2}
31 1−3T+pT2 1 - 3 T + p T^{2}
37 1−3T+pT2 1 - 3 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 1−2T+pT2 1 - 2 T + p T^{2}
53 1+T+pT2 1 + T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1−7T+pT2 1 - 7 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1+7T+pT2 1 + 7 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1+12T+pT2 1 + 12 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

−9.604100883180305119694719486737, −8.976631286949291998537604701701, −8.214855445870152436084038682621, −6.96290485918056996428743397373, −6.46925986382253407591418544135, −5.27646996011335233092203843669, −4.28393931774823338150480925867, −2.81267195278645779898796150454, −2.42545807821311073856045852532, 0, 2.42545807821311073856045852532, 2.81267195278645779898796150454, 4.28393931774823338150480925867, 5.27646996011335233092203843669, 6.46925986382253407591418544135, 6.96290485918056996428743397373, 8.214855445870152436084038682621, 8.976631286949291998537604701701, 9.604100883180305119694719486737

Graph of the ZZ-function along the critical line