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Properties

Label 4-1280e2-1.1-c3e2-0-25
Degree 44
Conductor 16384001638400
Sign 11
Analytic cond. 5703.635703.63
Root an. cond. 8.690368.69036
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·5-s − 44·9-s + 4·13-s + 92·17-s + 75·25-s + 232·29-s − 148·37-s − 480·41-s + 440·45-s − 436·49-s − 476·53-s + 1.35e3·61-s − 40·65-s − 1.95e3·73-s + 1.20e3·81-s − 920·85-s − 2.10e3·89-s + 548·97-s − 1.28e3·101-s + 580·109-s − 244·113-s − 176·117-s − 2.02e3·121-s − 500·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.62·9-s + 0.0853·13-s + 1.31·17-s + 3/5·25-s + 1.48·29-s − 0.657·37-s − 1.82·41-s + 1.45·45-s − 1.27·49-s − 1.23·53-s + 2.84·61-s − 0.0763·65-s − 3.13·73-s + 1.65·81-s − 1.17·85-s − 2.50·89-s + 0.573·97-s − 1.26·101-s + 0.509·109-s − 0.203·113-s − 0.139·117-s − 1.51·121-s − 0.357·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯

Functional equation

Λ(s)=(1638400s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1638400s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 16384001638400    =    216⋅522^{16} \cdot 5^{2}
Sign: 11
Analytic conductor: 5703.635703.63
Root analytic conductor: 8.690368.69036
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 1638400, ( :3/2,3/2), 1)(4,\ 1638400,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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+pT)2 ( 1 + p T )^{2}
good3C22C_2^2 1+44T2+p6T4 1 + 44 T^{2} + p^{6} T^{4}
7C22C_2^2 1+436T2+p6T4 1 + 436 T^{2} + p^{6} T^{4}
11C22C_2^2 1+2022T2+p6T4 1 + 2022 T^{2} + p^{6} T^{4}
13C2C_2 (1−2T+p3T2)2 ( 1 - 2 T + p^{3} T^{2} )^{2}
17C2C_2 (1−46T+p3T2)2 ( 1 - 46 T + p^{3} T^{2} )^{2}
19C22C_2^2 1−2p2T2+p6T4 1 - 2 p^{2} T^{2} + p^{6} T^{4}
23C22C_2^2 1−10476T2+p6T4 1 - 10476 T^{2} + p^{6} T^{4}
29C2C_2 (1−4pT+p3T2)2 ( 1 - 4 p T + p^{3} T^{2} )^{2}
31C22C_2^2 1+54742T2+p6T4 1 + 54742 T^{2} + p^{6} T^{4}
37C2C_2 (1+2pT+p3T2)2 ( 1 + 2 p T + p^{3} T^{2} )^{2}
41C2C_2 (1+240T+p3T2)2 ( 1 + 240 T + p^{3} T^{2} )^{2}
43C22C_2^2 1+108604T2+p6T4 1 + 108604 T^{2} + p^{6} T^{4}
47C22C_2^2 1+61236T2+p6T4 1 + 61236 T^{2} + p^{6} T^{4}
53C2C_2 (1+238T+p3T2)2 ( 1 + 238 T + p^{3} T^{2} )^{2}
59C22C_2^2 1+220318T2+p6T4 1 + 220318 T^{2} + p^{6} T^{4}
61C2C_2 (1−678T+p3T2)2 ( 1 - 678 T + p^{3} T^{2} )^{2}
67C22C_2^2 1+601436T2+p6T4 1 + 601436 T^{2} + p^{6} T^{4}
71C22C_2^2 1+240582T2+p6T4 1 + 240582 T^{2} + p^{6} T^{4}
73C2C_2 (1+978T+p3T2)2 ( 1 + 978 T + p^{3} T^{2} )^{2}
79C22C_2^2 1+886078T2+p6T4 1 + 886078 T^{2} + p^{6} T^{4}
83C22C_2^2 1+483084T2+p6T4 1 + 483084 T^{2} + p^{6} T^{4}
89C2C_2 (1+1050T+p3T2)2 ( 1 + 1050 T + p^{3} T^{2} )^{2}
97C2C_2 (1−274T+p3T2)2 ( 1 - 274 T + p^{3} 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

−8.865024397582155884948667792994, −8.558259978918684191814451251226, −8.319220103576333004935578660068, −8.080615308780335237249834979207, −7.47765800920758344060452704806, −7.13221283893254287337527364566, −6.43895243266082401476759860242, −6.40199972551993587911004738282, −5.54245764841009828813822625908, −5.39226870881797375594504846468, −4.91266688433564974900002609175, −4.37137720256668776793920583355, −3.71901149142790443334133210324, −3.34335628134583690205731754600, −2.92188866341407871154536718998, −2.53075107924500005423419044268, −1.53832001176797599123833877326, −1.04571775778523527492318985968, 0, 0, 1.04571775778523527492318985968, 1.53832001176797599123833877326, 2.53075107924500005423419044268, 2.92188866341407871154536718998, 3.34335628134583690205731754600, 3.71901149142790443334133210324, 4.37137720256668776793920583355, 4.91266688433564974900002609175, 5.39226870881797375594504846468, 5.54245764841009828813822625908, 6.40199972551993587911004738282, 6.43895243266082401476759860242, 7.13221283893254287337527364566, 7.47765800920758344060452704806, 8.080615308780335237249834979207, 8.319220103576333004935578660068, 8.558259978918684191814451251226, 8.865024397582155884948667792994

Graph of the ZZ-function along the critical line