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Properties

Label 4-9000e2-1.1-c1e2-0-12
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 6·11-s − 6·13-s + 3·17-s − 4·19-s − 2·23-s + 4·29-s − 5·31-s − 10·37-s + 41-s − 9·43-s − 10·47-s − 12·49-s + 5·53-s + 7·59-s − 11·61-s − 16·67-s + 13·71-s + 3·73-s + 6·77-s − 6·79-s − 13·83-s − 2·89-s − 6·91-s + 97-s + 20·101-s + 11·103-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.80·11-s − 1.66·13-s + 0.727·17-s − 0.917·19-s − 0.417·23-s + 0.742·29-s − 0.898·31-s − 1.64·37-s + 0.156·41-s − 1.37·43-s − 1.45·47-s − 1.71·49-s + 0.686·53-s + 0.911·59-s − 1.40·61-s − 1.95·67-s + 1.54·71-s + 0.351·73-s + 0.683·77-s − 0.675·79-s − 1.42·83-s − 0.211·89-s − 0.628·91-s + 0.101·97-s + 1.99·101-s + 1.08·103-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−T+13T2−pT3+p2T4 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4}
11C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
13C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
17D4D_{4} 1−3T+25T2−3pT3+p2T4 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4}
19D4D_{4} 1+4T+37T2+4pT3+p2T4 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4}
23C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
29D4D_{4} 1−4T+17T2−4pT3+p2T4 1 - 4 T + 17 T^{2} - 4 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+5T+67T2+5pT3+p2T4 1 + 5 T + 67 T^{2} + 5 p T^{3} + p^{2} T^{4}
37C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
41D4D_{4} 1−T+51T2−pT3+p2T4 1 - T + 51 T^{2} - p T^{3} + p^{2} T^{4}
43D4D_{4} 1+9T+95T2+9pT3+p2T4 1 + 9 T + 95 T^{2} + 9 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+10T+99T2+10pT3+p2T4 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−5T+11T2−5pT3+p2T4 1 - 5 T + 11 T^{2} - 5 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−7T+99T2−7pT3+p2T4 1 - 7 T + 99 T^{2} - 7 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+11T+91T2+11pT3+p2T4 1 + 11 T + 91 T^{2} + 11 p T^{3} + p^{2} T^{4}
67C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
71D4D_{4} 1−13T+173T2−13pT3+p2T4 1 - 13 T + 173 T^{2} - 13 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−3T+147T2−3pT3+p2T4 1 - 3 T + 147 T^{2} - 3 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+6T+42T2+6pT3+p2T4 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+13T+207T2+13pT3+p2T4 1 + 13 T + 207 T^{2} + 13 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+2T+159T2+2pT3+p2T4 1 + 2 T + 159 T^{2} + 2 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−T+193T2−pT3+p2T4 1 - T + 193 T^{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

−7.54726719309340410090036462955, −7.18095392503210580146057814504, −6.79867208305399667236234262571, −6.69843206989267076899089224253, −6.15387871654629026818727070313, −6.06135119365258827593464057962, −5.25821957376228107350716979989, −5.24086959334273367061854008256, −4.64905835335895814375447868681, −4.62890208332682623497980215809, −3.85361036268767644047246528884, −3.85300886583032049669113761785, −3.13709634540880602580262409944, −3.06687510260869426300241099772, −2.20606640864549811066135930900, −2.02221382474457516379643476012, −1.39710677739232928864602318780, −1.25292132931688424660773345651, 0, 0, 1.25292132931688424660773345651, 1.39710677739232928864602318780, 2.02221382474457516379643476012, 2.20606640864549811066135930900, 3.06687510260869426300241099772, 3.13709634540880602580262409944, 3.85300886583032049669113761785, 3.85361036268767644047246528884, 4.62890208332682623497980215809, 4.64905835335895814375447868681, 5.24086959334273367061854008256, 5.25821957376228107350716979989, 6.06135119365258827593464057962, 6.15387871654629026818727070313, 6.69843206989267076899089224253, 6.79867208305399667236234262571, 7.18095392503210580146057814504, 7.54726719309340410090036462955

Graph of the ZZ-function along the critical line