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Properties

Label 4-920e2-1.1-c1e2-0-17
Degree 44
Conductor 846400846400
Sign 11
Analytic cond. 53.967153.9671
Root an. cond. 2.710392.71039
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s − 7-s − 9-s − 4·11-s − 3·13-s + 2·15-s + 7·17-s − 4·19-s + 21-s − 2·23-s + 3·25-s − 8·29-s − 8·31-s + 4·33-s + 2·35-s − 3·37-s + 3·39-s − 4·43-s + 2·45-s − 5·47-s − 9·49-s − 7·51-s + 13·53-s + 8·55-s + 4·57-s − 7·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.377·7-s − 1/3·9-s − 1.20·11-s − 0.832·13-s + 0.516·15-s + 1.69·17-s − 0.917·19-s + 0.218·21-s − 0.417·23-s + 3/5·25-s − 1.48·29-s − 1.43·31-s + 0.696·33-s + 0.338·35-s − 0.493·37-s + 0.480·39-s − 0.609·43-s + 0.298·45-s − 0.729·47-s − 9/7·49-s − 0.980·51-s + 1.78·53-s + 1.07·55-s + 0.529·57-s − 0.911·59-s + ⋯

Functional equation

Λ(s)=(846400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 846400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(846400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 846400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 846400846400    =    26⋅52⋅2322^{6} \cdot 5^{2} \cdot 23^{2}
Sign: 11
Analytic conductor: 53.967153.9671
Root analytic conductor: 2.710392.71039
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 846400, ( :1/2,1/2), 1)(4,\ 846400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
5C1C_1 (1+T)2 ( 1 + T )^{2}
23C1C_1 (1+T)2 ( 1 + T )^{2}
good3D4D_{4} 1+T+2T2+pT3+p2T4 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4}
7D4D_{4} 1+T+10T2+pT3+p2T4 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4}
11C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
13D4D_{4} 1+3T+24T2+3pT3+p2T4 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−7T+42T2−7pT3+p2T4 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4}
19C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
29D4D_{4} 1+8T+57T2+8pT3+p2T4 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+8T+61T2+8pT3+p2T4 1 + 8 T + 61 T^{2} + 8 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+3T+72T2+3pT3+p2T4 1 + 3 T + 72 T^{2} + 3 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+65T2+p2T4 1 + 65 T^{2} + p^{2} T^{4}
43D4D_{4} 1+4T+22T2+4pT3+p2T4 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+5T+62T2+5pT3+p2T4 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−13T+144T2−13pT3+p2T4 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+7T+126T2+7pT3+p2T4 1 + 7 T + 126 T^{2} + 7 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+10T+130T2+10pT3+p2T4 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+19T+220T2+19pT3+p2T4 1 + 19 T + 220 T^{2} + 19 p T^{3} + p^{2} T^{4}
71C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
73D4D_{4} 1−11T+172T2−11pT3+p2T4 1 - 11 T + 172 T^{2} - 11 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+10T+166T2+10pT3+p2T4 1 + 10 T + 166 T^{2} + 10 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−3T+130T2−3pT3+p2T4 1 - 3 T + 130 T^{2} - 3 p T^{3} + p^{2} T^{4}
89C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
97D4D_{4} 1−2T+178T2−2pT3+p2T4 1 - 2 T + 178 T^{2} - 2 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

−10.04235762469178362002793081819, −9.527700492840389776085760445404, −8.918055005191489921962297743259, −8.744167355256539290499300744553, −7.87787094055281652610846232806, −7.82217264057467387909205067532, −7.41800561273076131734096444241, −7.08046810530948295305554100968, −6.26167348608412767415468072335, −6.00307995101330595635695902458, −5.32244973383612532078712401436, −5.25635349558868339763553177976, −4.58372953865440310153755007996, −3.99454410774820647075962821105, −3.34803918581233743638264399853, −3.11920822435374775943149076422, −2.27445453388986391326208563508, −1.54170528812686804589089880290, 0, 0, 1.54170528812686804589089880290, 2.27445453388986391326208563508, 3.11920822435374775943149076422, 3.34803918581233743638264399853, 3.99454410774820647075962821105, 4.58372953865440310153755007996, 5.25635349558868339763553177976, 5.32244973383612532078712401436, 6.00307995101330595635695902458, 6.26167348608412767415468072335, 7.08046810530948295305554100968, 7.41800561273076131734096444241, 7.82217264057467387909205067532, 7.87787094055281652610846232806, 8.744167355256539290499300744553, 8.918055005191489921962297743259, 9.527700492840389776085760445404, 10.04235762469178362002793081819

Graph of the ZZ-function along the critical line