↑

Properties

Label 4-9000e2-1.1-c1e2-0-14
Degree 44
Conductor 8100000081000000
Sign 11
Analytic cond. 5164.635164.63
Root an. cond. 8.477348.47734
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s + 2·11-s − 2·13-s − 3·17-s − 3·19-s + 3·23-s − 8·29-s − 5·31-s + 8·37-s + 6·41-s − 20·43-s + 9·47-s − 6·49-s − 53-s − 6·59-s − 11·61-s + 6·67-s − 14·71-s − 2·73-s + 4·77-s − 11·79-s + 83-s + 20·89-s − 4·91-s − 6·97-s + 6·101-s + 12·103-s + ⋯
L(s)  = 1  + 0.755·7-s + 0.603·11-s − 0.554·13-s − 0.727·17-s − 0.688·19-s + 0.625·23-s − 1.48·29-s − 0.898·31-s + 1.31·37-s + 0.937·41-s − 3.04·43-s + 1.31·47-s − 6/7·49-s − 0.137·53-s − 0.781·59-s − 1.40·61-s + 0.733·67-s − 1.66·71-s − 0.234·73-s + 0.455·77-s − 1.23·79-s + 0.109·83-s + 2.11·89-s − 0.419·91-s − 0.609·97-s + 0.597·101-s + 1.18·103-s + ⋯

Functional equation

Λ(s)=(81000000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 81000000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(81000000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81000000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8100000081000000    =    26⋅34⋅562^{6} \cdot 3^{4} \cdot 5^{6}
Sign: 11
Analytic conductor: 5164.635164.63
Root analytic conductor: 8.477348.47734
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 81000000, ( :1/2,1/2), 1)(4,\ 81000000,\ (\ :1/2, 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
5 1 1
good7D4D_{4} 1−2T+10T2−2pT3+p2T4 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4}
11D4D_{4} 1−2T+18T2−2pT3+p2T4 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4}
13D4D_{4} 1+2T+22T2+2pT3+p2T4 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+3T+35T2+3pT3+p2T4 1 + 3 T + 35 T^{2} + 3 p T^{3} + p^{2} T^{4}
19D4D_{4} 1+3T+39T2+3pT3+p2T4 1 + 3 T + 39 T^{2} + 3 p T^{3} + p^{2} T^{4}
23D4D_{4} 1−3T+47T2−3pT3+p2T4 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4}
29C4C_4 1+8T+54T2+8pT3+p2T4 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+5T+57T2+5pT3+p2T4 1 + 5 T + 57 T^{2} + 5 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−8T+70T2−8pT3+p2T4 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−6T+46T2−6pT3+p2T4 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4}
43C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
47D4D_{4} 1−9T+113T2−9pT3+p2T4 1 - 9 T + 113 T^{2} - 9 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+T+95T2+pT3+p2T4 1 + T + 95 T^{2} + p T^{3} + p^{2} T^{4}
59D4D_{4} 1+6T+122T2+6pT3+p2T4 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+11T+141T2+11pT3+p2T4 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−6T+138T2−6pT3+p2T4 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+14T+186T2+14pT3+p2T4 1 + 14 T + 186 T^{2} + 14 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+2T+102T2+2pT3+p2T4 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+11T+187T2+11pT3+p2T4 1 + 11 T + 187 T^{2} + 11 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−T+105T2−pT3+p2T4 1 - T + 105 T^{2} - p T^{3} + p^{2} T^{4}
89D4D_{4} 1−20T+258T2−20pT3+p2T4 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+6T+158T2+6pT3+p2T4 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4}
show more
show less
   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.54950892259714879953539379463, −7.35972990049802873612318575040, −6.84092419297923384874577721699, −6.55613449499843370710943959631, −6.22657661393673910892764474823, −5.94423965647449196866592912819, −5.37805617151707899110941103486, −5.17109985595309513323391077084, −4.71911471802450880915649858793, −4.50633739897342558499392459835, −4.00623068770582199970164609822, −3.80997504540090385577158650842, −3.18774796290811662273099437258, −2.91042644258290272161692912629, −2.17432288050964842618024054494, −2.13091268542081583844267243785, −1.34817922856538958334166454247, −1.27791120627764478213900858438, 0, 0, 1.27791120627764478213900858438, 1.34817922856538958334166454247, 2.13091268542081583844267243785, 2.17432288050964842618024054494, 2.91042644258290272161692912629, 3.18774796290811662273099437258, 3.80997504540090385577158650842, 4.00623068770582199970164609822, 4.50633739897342558499392459835, 4.71911471802450880915649858793, 5.17109985595309513323391077084, 5.37805617151707899110941103486, 5.94423965647449196866592912819, 6.22657661393673910892764474823, 6.55613449499843370710943959631, 6.84092419297923384874577721699, 7.35972990049802873612318575040, 7.54950892259714879953539379463

Graph of the ZZ-function along the critical line