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Properties

Label 4-540800-1.1-c1e2-0-18
Degree 44
Conductor 540800540800
Sign 11
Analytic cond. 34.481834.4818
Root an. cond. 2.423242.42324
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 4·7-s + 9-s + 2·13-s + 4·25-s − 12·35-s + 2·37-s − 3·45-s + 4·47-s − 49-s − 8·61-s + 4·63-s − 6·65-s + 16·67-s − 20·73-s + 8·79-s − 8·81-s + 24·83-s + 8·91-s + 8·97-s + 16·101-s + 2·117-s + 10·121-s + 3·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.51·7-s + 1/3·9-s + 0.554·13-s + 4/5·25-s − 2.02·35-s + 0.328·37-s − 0.447·45-s + 0.583·47-s − 1/7·49-s − 1.02·61-s + 0.503·63-s − 0.744·65-s + 1.95·67-s − 2.34·73-s + 0.900·79-s − 8/9·81-s + 2.63·83-s + 0.838·91-s + 0.812·97-s + 1.59·101-s + 0.184·117-s + 0.909·121-s + 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

Λ(s)=(540800s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(540800s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 540800540800    =    27⋅52⋅1322^{7} \cdot 5^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 34.481834.4818
Root analytic conductor: 2.423242.42324
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 540800, ( :1/2,1/2), 1)(4,\ 540800,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7985997791.798599779
L(12)L(\frac12) ≈\approx 1.7985997791.798599779
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
5C2C_2 1+3T+pT2 1 + 3 T + p T^{2}
13C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
good3C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
7C2C_2×\timesC2C_2 (1−3T+pT2)(1−T+pT2) ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} )
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
17C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
19C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
23C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
29C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
31C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
37C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
41C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
43C22C_2^2 1−9T2+p2T4 1 - 9 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−7T+pT2)(1+3T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} )
53C22C_2^2 1+82T2+p2T4 1 + 82 T^{2} + p^{2} T^{4}
59C22C_2^2 1−38T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−12T+pT2)(1−4T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} )
71C22C_2^2 1+55T2+p2T4 1 + 55 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1+6T+pT2)(1+14T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−14T+pT2)(1+6T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} )
83C2C_2×\timesC2C_2 (1−14T+pT2)(1−10T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} )
89C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{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

−8.455818670319071853073892193518, −7.968590726590330501123767305785, −7.64606590933620344425663891374, −7.32469371722216245826150993062, −6.76630124393435907765378699325, −6.17938178595195748928406873146, −5.67339607138908642684889275430, −5.00183036684168281129084284042, −4.59037596784972644267108292819, −4.31690762898032664126534932318, −3.60133389423961568180874434594, −3.24884191626243486637931941464, −2.25550588426683540942422150082, −1.60847664258435852494871434923, −0.74067720772506239042240082538, 0.74067720772506239042240082538, 1.60847664258435852494871434923, 2.25550588426683540942422150082, 3.24884191626243486637931941464, 3.60133389423961568180874434594, 4.31690762898032664126534932318, 4.59037596784972644267108292819, 5.00183036684168281129084284042, 5.67339607138908642684889275430, 6.17938178595195748928406873146, 6.76630124393435907765378699325, 7.32469371722216245826150993062, 7.64606590933620344425663891374, 7.968590726590330501123767305785, 8.455818670319071853073892193518

Graph of the ZZ-function along the critical line