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Properties

Label 4-800e2-1.1-c1e2-0-9
Degree 44
Conductor 640000640000
Sign 11
Analytic cond. 40.806940.8069
Root an. cond. 2.527452.52745
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)=(640000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(640000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 640000640000    =    210⋅542^{10} \cdot 5^{4}
Sign: 11
Analytic conductor: 40.806940.8069
Root analytic conductor: 2.527452.52745
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 640000, ( :1/2,1/2), 1)(4,\ 640000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7248888201.724888820
L(12)L(\frac12) ≈\approx 1.7248888201.724888820
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

−10.50605509965111304486465515300, −10.24993019668767040986526642586, −9.464476774997252509084814202742, −9.014783488310806066676906286092, −8.826822029803336769437210452676, −8.465430342484352186093050379282, −8.167905739342344370023067006366, −7.34755154133375962825895060082, −6.94760004129955496294405041358, −6.59803878243940616364742111278, −5.94426147354046117803981663122, −5.93614136368984664489504438907, −5.15500699103202102429843650560, −4.43230042203715345727770067725, −4.01699601747083075472287508810, −3.86599210376161220926146517499, −2.76239025809467826674546430899, −2.43843119291001336407519068539, −1.64894313034439815950593792217, −0.66921415445912516203127009829, 0.66921415445912516203127009829, 1.64894313034439815950593792217, 2.43843119291001336407519068539, 2.76239025809467826674546430899, 3.86599210376161220926146517499, 4.01699601747083075472287508810, 4.43230042203715345727770067725, 5.15500699103202102429843650560, 5.93614136368984664489504438907, 5.94426147354046117803981663122, 6.59803878243940616364742111278, 6.94760004129955496294405041358, 7.34755154133375962825895060082, 8.167905739342344370023067006366, 8.465430342484352186093050379282, 8.826822029803336769437210452676, 9.014783488310806066676906286092, 9.464476774997252509084814202742, 10.24993019668767040986526642586, 10.50605509965111304486465515300

Graph of the ZZ-function along the critical line