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Properties

Label 4-5408e2-1.1-c1e2-0-4
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 6·9-s − 2·17-s + 5·25-s + 10·29-s + 2·37-s − 10·41-s − 12·45-s − 14·49-s − 14·53-s + 10·61-s + 6·73-s + 27·81-s − 4·85-s + 20·89-s + 36·97-s + 2·101-s + 12·109-s + 14·113-s − 22·121-s + 22·125-s + 127-s + 131-s + 137-s + 139-s + 20·145-s + 149-s + ⋯
L(s)  = 1  + 0.894·5-s − 2·9-s − 0.485·17-s + 25-s + 1.85·29-s + 0.328·37-s − 1.56·41-s − 1.78·45-s − 2·49-s − 1.92·53-s + 1.28·61-s + 0.702·73-s + 3·81-s − 0.433·85-s + 2.11·89-s + 3.65·97-s + 0.199·101-s + 1.14·109-s + 1.31·113-s − 2·121-s + 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.66·145-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 2.1154417902.115441790
L(12)L(\frac12) ≈\approx 2.1154417902.115441790
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 1−2T−T2−2pT3+p2T4 1 - 2 T - T^{2} - 2 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 1−2T−33T2−2pT3+p2T4 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+10T+59T2+10pT3+p2T4 1 + 10 T + 59 T^{2} + 10 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 1−6T−37T2−6pT3+p2T4 1 - 6 T - 37 T^{2} - 6 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 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
97C2C_2 (1−18T+pT2)2 ( 1 - 18 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

−8.339404407806784979805063868463, −8.259940276924542474205691752400, −7.56251640150692743131899460682, −7.44269096659002166816521247633, −6.65470608501995748967457375573, −6.39600320238416325869614619073, −6.24545166216826244940788587369, −6.08506653508983395876665948111, −5.34041642192029349053559932620, −5.06024215683950327731809265843, −4.87616598719280955289602771171, −4.55691729694495304277263072565, −3.66054072176450290699381826535, −3.41617768225560963040881789265, −2.93410897743996099493501531590, −2.74152690375811162388719710257, −1.98012986105323017224554684664, −1.93205456336423180723752762190, −0.969544949121715982708019921159, −0.42824352894671353067376791343, 0.42824352894671353067376791343, 0.969544949121715982708019921159, 1.93205456336423180723752762190, 1.98012986105323017224554684664, 2.74152690375811162388719710257, 2.93410897743996099493501531590, 3.41617768225560963040881789265, 3.66054072176450290699381826535, 4.55691729694495304277263072565, 4.87616598719280955289602771171, 5.06024215683950327731809265843, 5.34041642192029349053559932620, 6.08506653508983395876665948111, 6.24545166216826244940788587369, 6.39600320238416325869614619073, 6.65470608501995748967457375573, 7.44269096659002166816521247633, 7.56251640150692743131899460682, 8.259940276924542474205691752400, 8.339404407806784979805063868463

Graph of the ZZ-function along the critical line