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Properties

Label 4-40e4-1.1-c1e2-0-36
Degree 44
Conductor 25600002560000
Sign 11
Analytic cond. 163.227163.227
Root an. cond. 3.574363.57436
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s + 8·13-s + 14·17-s + 4·37-s + 10·41-s + 6·49-s + 12·53-s − 20·61-s + 18·73-s − 8·81-s − 10·89-s − 4·97-s + 4·101-s − 12·109-s + 2·113-s − 8·117-s − 17·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 14·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1/3·9-s + 2.21·13-s + 3.39·17-s + 0.657·37-s + 1.56·41-s + 6/7·49-s + 1.64·53-s − 2.56·61-s + 2.10·73-s − 8/9·81-s − 1.05·89-s − 0.406·97-s + 0.398·101-s − 1.14·109-s + 0.188·113-s − 0.739·117-s − 1.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.13·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

Λ(s)=(2560000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2560000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 25600002560000    =    212⋅542^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 163.227163.227
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2560000, ( :1/2,1/2), 1)(4,\ 2560000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.4497776403.449777640
L(12)L(\frac12) ≈\approx 3.4497776403.449777640
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
5 1 1
good3C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
7C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
11C22C_2^2 1+17T2+p2T4 1 + 17 T^{2} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
17C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
19C22C_2^2 1−7T2+p2T4 1 - 7 T^{2} + p^{2} T^{4}
23C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C22C_2^2 1+42T2+p2T4 1 + 42 T^{2} + p^{2} T^{4}
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
43C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
47C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C22C_2^2 1+129T2+p2T4 1 + 129 T^{2} + p^{2} T^{4}
71C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
73C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
79C22C_2^2 1+138T2+p2T4 1 + 138 T^{2} + p^{2} T^{4}
83C22C_2^2 1+41T2+p2T4 1 + 41 T^{2} + p^{2} T^{4}
89C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
97C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
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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

−9.474626111714987005105452428289, −9.367359995447237594093830192117, −8.752116593395323242801923147869, −8.351852504600398419862760046948, −7.996825105458390711585015727874, −7.77361375731057492250159914064, −7.29576770062729789200049563035, −6.81597589739661095394337101429, −6.13240985028103065463642227541, −5.97216720530962660687726711371, −5.47532704787015703891955995501, −5.44232202253705255954242354875, −4.52793571040102260089164909665, −3.95258882913647270386748568644, −3.73742767373973948123899984573, −3.04418283685822809452542412477, −2.92892832482561331793376992509, −1.90743131121157376759686576618, −1.07252425096184911019067718479, −0.971631826011971749588523265304, 0.971631826011971749588523265304, 1.07252425096184911019067718479, 1.90743131121157376759686576618, 2.92892832482561331793376992509, 3.04418283685822809452542412477, 3.73742767373973948123899984573, 3.95258882913647270386748568644, 4.52793571040102260089164909665, 5.44232202253705255954242354875, 5.47532704787015703891955995501, 5.97216720530962660687726711371, 6.13240985028103065463642227541, 6.81597589739661095394337101429, 7.29576770062729789200049563035, 7.77361375731057492250159914064, 7.996825105458390711585015727874, 8.351852504600398419862760046948, 8.752116593395323242801923147869, 9.367359995447237594093830192117, 9.474626111714987005105452428289

Graph of the ZZ-function along the critical line