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Properties

Label 2-1440-1.1-c3-0-52
Degree 22
Conductor 14401440
Sign −1-1
Analytic cond. 84.962784.9627
Root an. cond. 9.217529.21752
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s + 6·7-s − 60·11-s + 50·13-s + 30·17-s + 40·19-s − 178·23-s + 25·25-s − 166·29-s + 20·31-s + 30·35-s + 10·37-s + 250·41-s + 142·43-s − 214·47-s − 307·49-s − 490·53-s − 300·55-s + 800·59-s + 250·61-s + 250·65-s − 774·67-s − 100·71-s − 230·73-s − 360·77-s − 1.32e3·79-s − 982·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.323·7-s − 1.64·11-s + 1.06·13-s + 0.428·17-s + 0.482·19-s − 1.61·23-s + 1/5·25-s − 1.06·29-s + 0.115·31-s + 0.144·35-s + 0.0444·37-s + 0.952·41-s + 0.503·43-s − 0.664·47-s − 0.895·49-s − 1.26·53-s − 0.735·55-s + 1.76·59-s + 0.524·61-s + 0.477·65-s − 1.41·67-s − 0.167·71-s − 0.368·73-s − 0.532·77-s − 1.87·79-s − 1.29·83-s + ⋯

Functional equation

Λ(s)=(1440s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1440s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14401440    =    25⋅32⋅52^{5} \cdot 3^{2} \cdot 5
Sign: −1-1
Analytic conductor: 84.962784.9627
Root analytic conductor: 9.217529.21752
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1440, ( :3/2), −1)(2,\ 1440,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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 1
5 1−pT 1 - p T
good7 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
11 1+60T+p3T2 1 + 60 T + p^{3} T^{2}
13 1−50T+p3T2 1 - 50 T + p^{3} T^{2}
17 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
19 1−40T+p3T2 1 - 40 T + p^{3} T^{2}
23 1+178T+p3T2 1 + 178 T + p^{3} T^{2}
29 1+166T+p3T2 1 + 166 T + p^{3} T^{2}
31 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
37 1−10T+p3T2 1 - 10 T + p^{3} T^{2}
41 1−250T+p3T2 1 - 250 T + p^{3} T^{2}
43 1−142T+p3T2 1 - 142 T + p^{3} T^{2}
47 1+214T+p3T2 1 + 214 T + p^{3} T^{2}
53 1+490T+p3T2 1 + 490 T + p^{3} T^{2}
59 1−800T+p3T2 1 - 800 T + p^{3} T^{2}
61 1−250T+p3T2 1 - 250 T + p^{3} T^{2}
67 1+774T+p3T2 1 + 774 T + p^{3} T^{2}
71 1+100T+p3T2 1 + 100 T + p^{3} T^{2}
73 1+230T+p3T2 1 + 230 T + p^{3} T^{2}
79 1+1320T+p3T2 1 + 1320 T + p^{3} T^{2}
83 1+982T+p3T2 1 + 982 T + p^{3} T^{2}
89 1+874T+p3T2 1 + 874 T + p^{3} T^{2}
97 1+310T+p3T2 1 + 310 T + p^{3} 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

−8.631110007440500643643641929041, −7.975291981124062710162686327239, −7.32040008714933576267079431405, −5.99884457062361662239430896150, −5.62401479097580832018562461052, −4.59655277249171470934803855051, −3.49836051256242586291057072221, −2.46129211346905869168313225818, −1.43201257363883792789469627823, 0, 1.43201257363883792789469627823, 2.46129211346905869168313225818, 3.49836051256242586291057072221, 4.59655277249171470934803855051, 5.62401479097580832018562461052, 5.99884457062361662239430896150, 7.32040008714933576267079431405, 7.975291981124062710162686327239, 8.631110007440500643643641929041

Graph of the ZZ-function along the critical line