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Properties

Label 2-4080-1.1-c1-0-51
Degree 22
Conductor 40804080
Sign −1-1
Analytic cond. 32.578932.5789
Root an. cond. 5.707795.70779
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 − 2·7-s + 9-s + 4·13-s − 15-s − 17-s − 4·19-s − 2·21-s − 4·23-s + 25-s + 27-s + 2·29-s + 2·35-s − 2·37-s + 4·39-s − 4·41-s − 10·43-s − 45-s + 8·47-s − 3·49-s − 51-s + 2·53-s − 4·57-s + 2·59-s − 14·61-s − 2·63-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.10·13-s − 0.258·15-s − 0.242·17-s − 0.917·19-s − 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.338·35-s − 0.328·37-s + 0.640·39-s − 0.624·41-s − 1.52·43-s − 0.149·45-s + 1.16·47-s − 3/7·49-s − 0.140·51-s + 0.274·53-s − 0.529·57-s + 0.260·59-s − 1.79·61-s − 0.251·63-s + ⋯

Functional equation

Λ(s)=(4080s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4080s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40804080    =    24⋅3⋅5⋅172^{4} \cdot 3 \cdot 5 \cdot 17
Sign: −1-1
Analytic conductor: 32.578932.5789
Root analytic conductor: 5.707795.70779
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4080, ( :1/2), −1)(2,\ 4080,\ (\ :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
3 1−T 1 - T
5 1+T 1 + T
17 1+T 1 + T
good7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+4T+pT2 1 + 4 T + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−2T+pT2 1 - 2 T + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1−12T+pT2 1 - 12 T + p T^{2}
83 1+8T+pT2 1 + 8 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1−8T+pT2 1 - 8 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

−8.257092370884583806553558157953, −7.38243293285553331487569791034, −6.54074197989563386558294722278, −6.10451359096155657339774956293, −4.94312421770451194001712015410, −3.99936858650052089229006018924, −3.50941212032853063313081131447, −2.58034758427836734985063034447, −1.48936096510746096838730928196, 0, 1.48936096510746096838730928196, 2.58034758427836734985063034447, 3.50941212032853063313081131447, 3.99936858650052089229006018924, 4.94312421770451194001712015410, 6.10451359096155657339774956293, 6.54074197989563386558294722278, 7.38243293285553331487569791034, 8.257092370884583806553558157953

Graph of the ZZ-function along the critical line