↑

Properties

Label 4-40e4-1.1-c1e2-0-32
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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5·9-s − 10·11-s + 10·19-s + 8·29-s + 20·31-s + 10·41-s + 10·49-s + 20·61-s − 20·79-s + 16·81-s + 18·89-s − 50·99-s − 4·101-s + 20·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 50·171-s + 173-s + ⋯
L(s)  = 1  + 5/3·9-s − 3.01·11-s + 2.29·19-s + 1.48·29-s + 3.59·31-s + 1.56·41-s + 10/7·49-s + 2.56·61-s − 2.25·79-s + 16/9·81-s + 1.90·89-s − 5.02·99-s − 0.398·101-s + 1.91·109-s + 4.81·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 3.82·171-s + 0.0760·173-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 2.9294332932.929433293
L(12)L(\frac12) ≈\approx 2.9294332932.929433293
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 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
13C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C22C_2^2 1−9T2+p2T4 1 - 9 T^{2} + p^{2} T^{4}
19C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
23C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
29C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
31C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
37C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
41C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
53C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
67C22C_2^2 1−125T2+p2T4 1 - 125 T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1−121T2+p2T4 1 - 121 T^{2} + p^{2} T^{4}
79C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
83C22C_2^2 1−165T2+p2T4 1 - 165 T^{2} + p^{2} T^{4}
89C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
97C22C_2^2 1−94T2+p2T4 1 - 94 T^{2} + 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

−9.885719314496483631087440279456, −9.444920249324078363851924348847, −8.489964967267012127252391952130, −8.451591241654617797928958535046, −8.011790149183488762133875175113, −7.48827396405676716769625049755, −7.31292880784136462291527510761, −7.08781247594143050208155581985, −6.27821204934432876523972304711, −5.99446293720170483428410644915, −5.34149140931135853662442739545, −5.09053976100896970407975199078, −4.54194332334893634143718979972, −4.47884914177644198254298165904, −3.57852601168250932625819451055, −3.00848023143031987805206135530, −2.53837244369318263664053504632, −2.33720387336688582995098144612, −0.988459762065002097456050266730, −0.899040366512349458854508529172, 0.899040366512349458854508529172, 0.988459762065002097456050266730, 2.33720387336688582995098144612, 2.53837244369318263664053504632, 3.00848023143031987805206135530, 3.57852601168250932625819451055, 4.47884914177644198254298165904, 4.54194332334893634143718979972, 5.09053976100896970407975199078, 5.34149140931135853662442739545, 5.99446293720170483428410644915, 6.27821204934432876523972304711, 7.08781247594143050208155581985, 7.31292880784136462291527510761, 7.48827396405676716769625049755, 8.011790149183488762133875175113, 8.451591241654617797928958535046, 8.489964967267012127252391952130, 9.444920249324078363851924348847, 9.885719314496483631087440279456

Graph of the ZZ-function along the critical line