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Properties

Label 4-90e4-1.1-c1e2-0-15
Degree 44
Conductor 6561000065610000
Sign 11
Analytic cond. 4183.354183.35
Root an. cond. 8.042318.04231
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 4·13-s − 6·17-s − 2·19-s + 12·29-s − 2·31-s + 2·37-s + 12·41-s − 10·43-s − 6·47-s − 8·49-s − 18·53-s + 12·59-s − 8·61-s + 8·67-s + 12·71-s − 10·73-s − 8·79-s − 6·83-s − 8·91-s + 2·97-s + 12·101-s − 16·103-s − 2·109-s − 18·113-s − 12·119-s − 19·121-s + ⋯
L(s)  = 1  + 0.755·7-s − 1.10·13-s − 1.45·17-s − 0.458·19-s + 2.22·29-s − 0.359·31-s + 0.328·37-s + 1.87·41-s − 1.52·43-s − 0.875·47-s − 8/7·49-s − 2.47·53-s + 1.56·59-s − 1.02·61-s + 0.977·67-s + 1.42·71-s − 1.17·73-s − 0.900·79-s − 0.658·83-s − 0.838·91-s + 0.203·97-s + 1.19·101-s − 1.57·103-s − 0.191·109-s − 1.69·113-s − 1.10·119-s − 1.72·121-s + ⋯

Functional equation

Λ(s)=(65610000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(65610000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 6561000065610000    =    24⋅38⋅542^{4} \cdot 3^{8} \cdot 5^{4}
Sign: 11
Analytic conductor: 4183.354183.35
Root analytic conductor: 8.042318.04231
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 65610000, ( :1/2,1/2), 1)(4,\ 65610000,\ (\ :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+12T2−2pT3+p2T4 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+19T2+p2T4 1 + 19 T^{2} + p^{2} T^{4}
13D4D_{4} 1+4T+18T2+4pT3+p2T4 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+6T+40T2+6pT3+p2T4 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4}
19D4D_{4} 1+2T+27T2+2pT3+p2T4 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
29D4D_{4} 1−12T+91T2−12pT3+p2T4 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+2T+15T2+2pT3+p2T4 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−2T+48T2−2pT3+p2T4 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−12T+91T2−12pT3+p2T4 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+10T+108T2+10pT3+p2T4 1 + 10 T + 108 T^{2} + 10 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+6T+100T2+6pT3+p2T4 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+18T+184T2+18pT3+p2T4 1 + 18 T + 184 T^{2} + 18 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−12T+151T2−12pT3+p2T4 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4}
61C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
67D4D_{4} 1−8T+42T2−8pT3+p2T4 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−12T+151T2−12pT3+p2T4 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+10T+144T2+10pT3+p2T4 1 + 10 T + 144 T^{2} + 10 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+8T+66T2+8pT3+p2T4 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+6T+28T2+6pT3+p2T4 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+151T2+p2T4 1 + 151 T^{2} + p^{2} T^{4}
97D4D_{4} 1−2T+192T2−2pT3+p2T4 1 - 2 T + 192 T^{2} - 2 p T^{3} + p^{2} T^{4}
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   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

−7.52823112670035452477514657938, −7.51284282937636357077512343951, −6.83311705257973475996541678650, −6.61797779700303150396046229557, −6.30170526659815313556525023665, −6.17407753021035478280464558705, −5.34575602940067639386150980585, −5.17677881684802930378837505420, −4.72091041102345550663197415681, −4.61434920652647565414799601207, −4.26832180120372390562998740704, −3.72626857043563210723674310832, −3.29510237553682378587951643323, −2.67166675170214527963382748820, −2.49163737578087806360969116730, −2.14842252761752349386239190740, −1.31218399016290879851262057647, −1.28578020329537963669087552742, 0, 0, 1.28578020329537963669087552742, 1.31218399016290879851262057647, 2.14842252761752349386239190740, 2.49163737578087806360969116730, 2.67166675170214527963382748820, 3.29510237553682378587951643323, 3.72626857043563210723674310832, 4.26832180120372390562998740704, 4.61434920652647565414799601207, 4.72091041102345550663197415681, 5.17677881684802930378837505420, 5.34575602940067639386150980585, 6.17407753021035478280464558705, 6.30170526659815313556525023665, 6.61797779700303150396046229557, 6.83311705257973475996541678650, 7.51284282937636357077512343951, 7.52823112670035452477514657938

Graph of the ZZ-function along the critical line