↑

Properties

Label 4-540800-1.1-c1e2-0-31
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·9-s + 2·13-s + 4·17-s + 3·25-s − 4·29-s − 4·37-s + 4·41-s + 4·45-s + 2·49-s + 12·53-s − 4·61-s + 4·65-s + 4·73-s − 5·81-s + 8·85-s + 4·89-s − 12·97-s + 12·101-s + 12·109-s − 12·113-s + 4·117-s + 18·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s + 2/3·9-s + 0.554·13-s + 0.970·17-s + 3/5·25-s − 0.742·29-s − 0.657·37-s + 0.624·41-s + 0.596·45-s + 2/7·49-s + 1.64·53-s − 0.512·61-s + 0.496·65-s + 0.468·73-s − 5/9·81-s + 0.867·85-s + 0.423·89-s − 1.21·97-s + 1.19·101-s + 1.14·109-s − 1.12·113-s + 0.369·117-s + 1.63·121-s + 0.357·125-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 2.6950357142.695035714
L(12)L(\frac12) ≈\approx 2.6950357142.695035714
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}
13C1C_1 (1−T)2 ( 1 - T )^{2}
good3C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
23C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
29C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C22C_2^2 1−42T2+p2T4 1 - 42 T^{2} + p^{2} T^{4}
37C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
41C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
47C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
71C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
73C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
79C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
83C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−2T+pT2)(1+14T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} )
show more
show less
   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

−8.546465749511268910415782865259, −7.987109424737057245307656397071, −7.53125199949556523939117458653, −7.09717057872848487510621024148, −6.68862159372961218064790446970, −6.08902705980528534510708629881, −5.67575443421577017654436013888, −5.37529001107628665578409038673, −4.71703251013746342009376808605, −4.13991613053845340278766409570, −3.60795635925484158023154455185, −3.03540755191807680960141033038, −2.25957846055533507593722255797, −1.65967151690953797211524541958, −0.919009355606882619009403181064, 0.919009355606882619009403181064, 1.65967151690953797211524541958, 2.25957846055533507593722255797, 3.03540755191807680960141033038, 3.60795635925484158023154455185, 4.13991613053845340278766409570, 4.71703251013746342009376808605, 5.37529001107628665578409038673, 5.67575443421577017654436013888, 6.08902705980528534510708629881, 6.68862159372961218064790446970, 7.09717057872848487510621024148, 7.53125199949556523939117458653, 7.987109424737057245307656397071, 8.546465749511268910415782865259

Graph of the ZZ-function along the critical line