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Properties

Label 4-5408e2-1.1-c1e2-0-3
Degree 44
Conductor 2924646429246464
Sign 11
Analytic cond. 1864.771864.77
Root an. cond. 6.571386.57138
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 6·9-s − 2·17-s + 5·25-s − 10·29-s + 12·37-s − 8·41-s + 24·45-s − 14·49-s − 14·53-s + 10·61-s − 16·73-s + 27·81-s + 8·85-s + 32·89-s + 16·97-s − 2·101-s + 40·109-s − 14·113-s − 22·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 40·145-s + 149-s + ⋯
L(s)  = 1  − 1.78·5-s − 2·9-s − 0.485·17-s + 25-s − 1.85·29-s + 1.97·37-s − 1.24·41-s + 3.57·45-s − 2·49-s − 1.92·53-s + 1.28·61-s − 1.87·73-s + 3·81-s + 0.867·85-s + 3.39·89-s + 1.62·97-s − 0.199·101-s + 3.83·109-s − 1.31·113-s − 2·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.32·145-s + 0.0819·149-s + ⋯

Functional equation

Λ(s)=(29246464s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(29246464s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2924646429246464    =    210⋅1342^{10} \cdot 13^{4}
Sign: 11
Analytic conductor: 1864.771864.77
Root analytic conductor: 6.571386.57138
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 29246464, ( :1/2,1/2), 1)(4,\ 29246464,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.58671798800.5867179880
L(12)L(\frac12) ≈\approx 0.58671798800.5867179880
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
13 1 1
good3C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
5C22C_2^2 1+4T+11T2+4pT3+p2T4 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4}
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C22C_2^2 1+2T−13T2+2pT3+p2T4 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4}
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C22C_2^2 1+10T+71T2+10pT3+p2T4 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C22C_2^2 1−12T+107T2−12pT3+p2T4 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+8T+23T2+8pT3+p2T4 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4}
43C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C22C_2^2 1+14T+143T2+14pT3+p2T4 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C22C_2^2 1−10T+39T2−10pT3+p2T4 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4}
67C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1+16T+183T2+16pT3+p2T4 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}
97C2C_2 (1−8T+pT2)2 ( 1 - 8 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

−8.047267363032367249238689036218, −8.009443536192604823779096498288, −7.70276162878071649176394814594, −7.66454290419827783715394538691, −6.76154468922796015742242014365, −6.71140093837909891543444052754, −6.18626943713450779423155043873, −5.84596837556311646541770333038, −5.48831457194904525585714366812, −5.06271233566061882317080150927, −4.49591275935552160989500507141, −4.47767317041287295168331492622, −3.67080431120527926069607542902, −3.57625342560398635037056326806, −3.02643055368253838528120263197, −2.93338382182881838519913474230, −1.95192677058092351707406748068, −1.91554975604067973069524470670, −0.66691342743865919612805953123, −0.30897783444820330366712468730, 0.30897783444820330366712468730, 0.66691342743865919612805953123, 1.91554975604067973069524470670, 1.95192677058092351707406748068, 2.93338382182881838519913474230, 3.02643055368253838528120263197, 3.57625342560398635037056326806, 3.67080431120527926069607542902, 4.47767317041287295168331492622, 4.49591275935552160989500507141, 5.06271233566061882317080150927, 5.48831457194904525585714366812, 5.84596837556311646541770333038, 6.18626943713450779423155043873, 6.71140093837909891543444052754, 6.76154468922796015742242014365, 7.66454290419827783715394538691, 7.70276162878071649176394814594, 8.009443536192604823779096498288, 8.047267363032367249238689036218

Graph of the ZZ-function along the critical line