↑

Properties

Label 4-9702e2-1.1-c1e2-0-1
Degree 44
Conductor 9412880494128804
Sign 11
Analytic cond. 6001.736001.73
Root an. cond. 8.801758.80175
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  − 2·2-s + 3·4-s − 2·5-s − 4·8-s + 4·10-s + 2·11-s + 5·16-s − 6·17-s + 2·19-s − 6·20-s − 4·22-s − 4·23-s − 2·25-s + 10·31-s − 6·32-s + 12·34-s − 4·37-s − 4·38-s + 8·40-s − 6·41-s + 4·43-s + 6·44-s + 8·46-s − 6·47-s + 4·50-s + 12·53-s − 4·55-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.41·8-s + 1.26·10-s + 0.603·11-s + 5/4·16-s − 1.45·17-s + 0.458·19-s − 1.34·20-s − 0.852·22-s − 0.834·23-s − 2/5·25-s + 1.79·31-s − 1.06·32-s + 2.05·34-s − 0.657·37-s − 0.648·38-s + 1.26·40-s − 0.937·41-s + 0.609·43-s + 0.904·44-s + 1.17·46-s − 0.875·47-s + 0.565·50-s + 1.64·53-s − 0.539·55-s + ⋯

Functional equation

Λ(s)=(94128804s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(94128804s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9412880494128804    =    22⋅34⋅74⋅1122^{2} \cdot 3^{4} \cdot 7^{4} \cdot 11^{2}
Sign: 11
Analytic conductor: 6001.736001.73
Root analytic conductor: 8.801758.80175
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 94128804, ( :1/2,1/2), 1)(4,\ 94128804,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.86389137440.8638913744
L(12)L(\frac12) ≈\approx 0.86389137440.8638913744
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)
bad2C1C_1 (1+T)2 ( 1 + T )^{2}
3 1 1
7 1 1
11C1C_1 (1−T)2 ( 1 - T )^{2}
good5D4D_{4} 1+2T+6T2+2pT3+p2T4 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17D4D_{4} 1+6T+38T2+6pT3+p2T4 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4}
19C4C_4 1−2T−6T2−2pT3+p2T4 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+4T+30T2+4pT3+p2T4 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
31D4D_{4} 1−10T+82T2−10pT3+p2T4 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+4T−2T2+4pT3+p2T4 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+6T+86T2+6pT3+p2T4 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−4T+70T2−4pT3+p2T4 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+6T+58T2+6pT3+p2T4 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59D4D_{4} 1+12T+134T2+12pT3+p2T4 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4}
61C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
67D4D_{4} 1−4T−42T2−4pT3+p2T4 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4}
71C4C_4 1+4T+126T2+4pT3+p2T4 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−18T+182T2−18pT3+p2T4 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+4T−18T2+4pT3+p2T4 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+2T+162T2+2pT3+p2T4 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+4T+162T2+4pT3+p2T4 1 + 4 T + 162 T^{2} + 4 p T^{3} + p^{2} T^{4}
97C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
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

−7.84765487368768860769242742653, −7.71992016479634089860880991159, −7.15020599873679668348541086871, −6.90951080462553017174952159381, −6.62506979527121883220720456246, −6.43818126420135066311204821204, −5.93481120913551524054522139116, −5.54100819292567442174291173581, −5.23341744295641218995391737590, −4.51043541353326318563307519364, −4.35898993130214179007988660286, −4.07505504720581808236900605838, −3.37851601074741963049161761425, −3.32548423912170175310326166844, −2.59968979450742003780057019075, −2.37519692820945728285819048018, −1.69048504214662511733193828091, −1.54866508644472456882636244626, −0.67258874766466695522463288606, −0.39589012665285907675282373958, 0.39589012665285907675282373958, 0.67258874766466695522463288606, 1.54866508644472456882636244626, 1.69048504214662511733193828091, 2.37519692820945728285819048018, 2.59968979450742003780057019075, 3.32548423912170175310326166844, 3.37851601074741963049161761425, 4.07505504720581808236900605838, 4.35898993130214179007988660286, 4.51043541353326318563307519364, 5.23341744295641218995391737590, 5.54100819292567442174291173581, 5.93481120913551524054522139116, 6.43818126420135066311204821204, 6.62506979527121883220720456246, 6.90951080462553017174952159381, 7.15020599873679668348541086871, 7.71992016479634089860880991159, 7.84765487368768860769242742653

Graph of the ZZ-function along the critical line