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Properties

Label 4-540800-1.1-c1e2-0-24
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  − 2-s + 4-s − 5-s + 4·7-s − 8-s + 9-s + 10-s + 6·13-s − 4·14-s + 16-s − 18-s − 20-s − 4·25-s − 6·26-s + 4·28-s + 8·29-s − 32-s − 4·35-s + 36-s − 10·37-s + 40-s − 45-s + 4·47-s + 7·49-s + 4·50-s + 6·52-s − 4·56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.66·13-s − 1.06·14-s + 1/4·16-s − 0.235·18-s − 0.223·20-s − 4/5·25-s − 1.17·26-s + 0.755·28-s + 1.48·29-s − 0.176·32-s − 0.676·35-s + 1/6·36-s − 1.64·37-s + 0.158·40-s − 0.149·45-s + 0.583·47-s + 49-s + 0.565·50-s + 0.832·52-s − 0.534·56-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 1.7835770511.783577051
L(12)L(\frac12) ≈\approx 1.7835770511.783577051
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)
bad2C1C_1 1+T 1 + T
5C2C_2 1+T+pT2 1 + T + p T^{2}
13C2C_2 1−6T+pT2 1 - 6 T + p T^{2}
good3C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
7C2C_2 (1−5T+pT2)(1+T+pT2) ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} )
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
19C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
23C22C_2^2 1−38T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
29C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
31C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
37C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C22C_2^2 1+39T2+p2T4 1 + 39 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−9T+pT2)(1+5T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 5 T + p T^{2} )
53C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
59C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
67C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
71C22C_2^2 1+47T2+p2T4 1 + 47 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−14T+pT2)(1−10T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} )
79C2C_2×\timesC2C_2 (1+2T+pT2)(1+10T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} )
83C2C_2×\timesC2C_2 (1−18T+pT2)(1+6T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} )
89C22C_2^2 1+82T2+p2T4 1 + 82 T^{2} + p^{2} T^{4}
97C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p 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

−8.441478766003148556623796299623, −8.144095561645493793928142696150, −7.79553307742508526649912506501, −7.15207845955938326404734101537, −6.77144560653655953790652734450, −6.33155021821359964214935134347, −5.66369001367489491921436286434, −5.24767560053836964503588703269, −4.72221612142743047282768708968, −3.97326277707734270004818295514, −3.79580057738805113963063133049, −2.95435702019764121095310342586, −2.10228754368625768564730352600, −1.53550791444121486048744677720, −0.856906510162455468474606832055, 0.856906510162455468474606832055, 1.53550791444121486048744677720, 2.10228754368625768564730352600, 2.95435702019764121095310342586, 3.79580057738805113963063133049, 3.97326277707734270004818295514, 4.72221612142743047282768708968, 5.24767560053836964503588703269, 5.66369001367489491921436286434, 6.33155021821359964214935134347, 6.77144560653655953790652734450, 7.15207845955938326404734101537, 7.79553307742508526649912506501, 8.144095561645493793928142696150, 8.441478766003148556623796299623

Graph of the ZZ-function along the critical line