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Properties

Label 4-700e2-1.1-c5e2-0-3
Degree 44
Conductor 490000490000
Sign 11
Analytic cond. 12604.212604.2
Root an. cond. 10.595610.5956
Motivic weight 55
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  − 190·9-s + 16·11-s + 5.39e3·19-s + 6.50e3·29-s + 9.57e3·31-s + 2.67e4·41-s − 2.40e3·49-s − 6.94e4·59-s − 2.06e3·61-s + 1.25e5·71-s − 2.28e4·79-s − 2.29e4·81-s − 3.94e4·89-s − 3.04e3·99-s + 9.18e4·101-s + 2.92e5·109-s − 3.21e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.74e5·169-s + ⋯
L(s)  = 1  − 0.781·9-s + 0.0398·11-s + 3.42·19-s + 1.43·29-s + 1.78·31-s + 2.48·41-s − 1/7·49-s − 2.59·59-s − 0.0710·61-s + 2.95·71-s − 0.411·79-s − 0.388·81-s − 0.527·89-s − 0.0311·99-s + 0.895·101-s + 2.36·109-s − 1.99·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.739·169-s + ⋯

Functional equation

Λ(s)=(490000s/2 ΓC(s)2 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(490000s/2 ΓC(s+5/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 490000490000    =    24⋅54⋅722^{4} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 12604.212604.2
Root analytic conductor: 10.595610.5956
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 490000, ( :5/2,5/2), 1)(4,\ 490000,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 4.7869684944.786968494
L(12)L(\frac12) ≈\approx 4.7869684944.786968494
L(72)L(\frac{7}{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
5 1 1
7C2C_2 1+p4T2 1 + p^{4} T^{2}
good3C22C_2^2 1+190T2+p10T4 1 + 190 T^{2} + p^{10} T^{4}
11C2C_2 (1−8T+p5T2)2 ( 1 - 8 T + p^{5} T^{2} )^{2}
13C22C_2^2 1−274730T2+p10T4 1 - 274730 T^{2} + p^{10} T^{4}
17C22C_2^2 1+2079810T2+p10T4 1 + 2079810 T^{2} + p^{10} T^{4}
19C2C_2 (1−142pT+p5T2)2 ( 1 - 142 p T + p^{5} T^{2} )^{2}
23C22C_2^2 1−1690350T2+p10T4 1 - 1690350 T^{2} + p^{10} T^{4}
29C2C_2 (1−3254T+p5T2)2 ( 1 - 3254 T + p^{5} T^{2} )^{2}
31C2C_2 (1−4788T+p5T2)2 ( 1 - 4788 T + p^{5} T^{2} )^{2}
37C22C_2^2 1−5206p2T2+p10T4 1 - 5206 p^{2} T^{2} + p^{10} T^{4}
41C2C_2 (1−13350T+p5T2)2 ( 1 - 13350 T + p^{5} T^{2} )^{2}
43C22C_2^2 1−293155702T2+p10T4 1 - 293155702 T^{2} + p^{10} T^{4}
47C22C_2^2 1−457221070T2+p10T4 1 - 457221070 T^{2} + p^{10} T^{4}
53C22C_2^2 1−664518886T2+p10T4 1 - 664518886 T^{2} + p^{10} T^{4}
59C2C_2 (1+34702T+p5T2)2 ( 1 + 34702 T + p^{5} T^{2} )^{2}
61C2C_2 (1+1032T+p5T2)2 ( 1 + 1032 T + p^{5} T^{2} )^{2}
67C22C_2^2 1−2598078550T2+p10T4 1 - 2598078550 T^{2} + p^{10} T^{4}
71C2C_2 (1−62720T+p5T2)2 ( 1 - 62720 T + p^{5} T^{2} )^{2}
73C22C_2^2 1−3787949710T2+p10T4 1 - 3787949710 T^{2} + p^{10} T^{4}
79C2C_2 (1+11400T+p5T2)2 ( 1 + 11400 T + p^{5} T^{2} )^{2}
83C22C_2^2 1+35444478T2+p10T4 1 + 35444478 T^{2} + p^{10} T^{4}
89C2C_2 (1+19722T+p5T2)2 ( 1 + 19722 T + p^{5} T^{2} )^{2}
97C22C_2^2 1−16883568670T2+p10T4 1 - 16883568670 T^{2} + p^{10} 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

−9.978618610704469812879841866272, −9.508189959917088312837977440987, −9.041343905110711834432064447421, −8.728276621716262929981221106362, −7.943532404469563784487236275034, −7.81210433870065799153423810838, −7.48228842517897473000182117728, −6.74447270860215779604596003012, −6.39585942779594544159095719289, −5.78185358935566345258333302199, −5.53480715811095869125605992734, −4.82973845970495368590942040896, −4.63685170583588578708517041723, −3.82848343290785251648986693058, −3.10962668352713938753962872717, −2.96721600251874651710041593643, −2.42662523741341797727763572820, −1.43653113787135412477803022112, −0.877851811783822040198685488929, −0.61082719206203122898371408343, 0.61082719206203122898371408343, 0.877851811783822040198685488929, 1.43653113787135412477803022112, 2.42662523741341797727763572820, 2.96721600251874651710041593643, 3.10962668352713938753962872717, 3.82848343290785251648986693058, 4.63685170583588578708517041723, 4.82973845970495368590942040896, 5.53480715811095869125605992734, 5.78185358935566345258333302199, 6.39585942779594544159095719289, 6.74447270860215779604596003012, 7.48228842517897473000182117728, 7.81210433870065799153423810838, 7.943532404469563784487236275034, 8.728276621716262929981221106362, 9.041343905110711834432064447421, 9.508189959917088312837977440987, 9.978618610704469812879841866272

Graph of the ZZ-function along the critical line