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Properties

Label 4-364e2-1.1-c1e2-0-9
Degree 44
Conductor 132496132496
Sign 11
Analytic cond. 8.448058.44805
Root an. cond. 1.704861.70486
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  + 2·3-s + 2·7-s + 6·11-s + 2·13-s + 6·17-s + 4·19-s + 4·21-s + 6·23-s − 7·25-s − 2·27-s − 6·29-s − 8·31-s + 12·33-s − 2·37-s + 4·39-s − 14·43-s + 12·47-s + 3·49-s + 12·51-s − 6·53-s + 8·57-s − 20·61-s − 8·67-s + 12·69-s − 6·71-s − 8·73-s − 14·75-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s + 1.80·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.872·21-s + 1.25·23-s − 7/5·25-s − 0.384·27-s − 1.11·29-s − 1.43·31-s + 2.08·33-s − 0.328·37-s + 0.640·39-s − 2.13·43-s + 1.75·47-s + 3/7·49-s + 1.68·51-s − 0.824·53-s + 1.05·57-s − 2.56·61-s − 0.977·67-s + 1.44·69-s − 0.712·71-s − 0.936·73-s − 1.61·75-s + ⋯

Functional equation

Λ(s)=(132496s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(132496s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 132496132496    =    24⋅72⋅1322^{4} \cdot 7^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 8.448058.44805
Root analytic conductor: 1.704861.70486
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 132496, ( :1/2,1/2), 1)(4,\ 132496,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.9299345282.929934528
L(12)L(\frac12) ≈\approx 2.9299345282.929934528
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
7C1C_1 (1−T)2 ( 1 - T )^{2}
13C1C_1 (1−T)2 ( 1 - T )^{2}
good3D4D_{4} 1−2T+4T2−2pT3+p2T4 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4}
5C22C_2^2 1+7T2+p2T4 1 + 7 T^{2} + p^{2} T^{4}
11D4D_{4} 1−6T+28T2−6pT3+p2T4 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−6T+16T2−6pT3+p2T4 1 - 6 T + 16 T^{2} - 6 p T^{3} + p^{2} T^{4}
19D4D_{4} 1−4T+15T2−4pT3+p2T4 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4}
23C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
29D4D_{4} 1+6T+55T2+6pT3+p2T4 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+8T+51T2+8pT3+p2T4 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+2T+48T2+2pT3+p2T4 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
43C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
47D4D_{4} 1−12T+127T2−12pT3+p2T4 1 - 12 T + 127 T^{2} - 12 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+6T+67T2+6pT3+p2T4 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67D4D_{4} 1+8T+42T2+8pT3+p2T4 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+6T+148T2+6pT3+p2T4 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+8T+135T2+8pT3+p2T4 1 + 8 T + 135 T^{2} + 8 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+2T+51T2+2pT3+p2T4 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+12T+175T2+12pT3+p2T4 1 + 12 T + 175 T^{2} + 12 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−12T+187T2−12pT3+p2T4 1 - 12 T + 187 T^{2} - 12 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−16T+231T2−16pT3+p2T4 1 - 16 T + 231 T^{2} - 16 p T^{3} + p^{2} T^{4}
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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

−11.55354777303285183537828862048, −11.48162942371729126931923445671, −10.62562328018689743867872336853, −10.36084466657607284142720822721, −9.427953620658775691068582805548, −9.343634114620225412128553460420, −8.919002538328781227320795980179, −8.561716837758046088552720669195, −7.80985016415399555533128753352, −7.60300856211437116905583953285, −7.14403758966548007283223252676, −6.39449945158992490277324879979, −5.73070678677033449952453341354, −5.44610844428662567205651774747, −4.58578199167383713369794242349, −3.90423520744475782645598411607, −3.35473243646736378058596595982, −3.08037979132238109847459385699, −1.76859295600357545086886342204, −1.40394088187422053685394686066, 1.40394088187422053685394686066, 1.76859295600357545086886342204, 3.08037979132238109847459385699, 3.35473243646736378058596595982, 3.90423520744475782645598411607, 4.58578199167383713369794242349, 5.44610844428662567205651774747, 5.73070678677033449952453341354, 6.39449945158992490277324879979, 7.14403758966548007283223252676, 7.60300856211437116905583953285, 7.80985016415399555533128753352, 8.561716837758046088552720669195, 8.919002538328781227320795980179, 9.343634114620225412128553460420, 9.427953620658775691068582805548, 10.36084466657607284142720822721, 10.62562328018689743867872336853, 11.48162942371729126931923445671, 11.55354777303285183537828862048

Graph of the ZZ-function along the critical line