↑

Properties

Label 4-259200-1.1-c1e2-0-28
Degree 44
Conductor 259200259200
Sign 11
Analytic cond. 16.526816.5268
Root an. cond. 2.016262.01626
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-s + 4-s + 4·5-s − 8-s − 4·10-s + 16-s − 2·19-s + 4·20-s − 2·23-s + 11·25-s + 4·29-s − 32-s + 2·38-s − 4·40-s + 4·43-s + 2·46-s + 12·47-s + 4·49-s − 11·50-s + 16·53-s − 4·58-s + 64-s − 10·67-s − 2·71-s − 4·73-s − 2·76-s + 4·80-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.353·8-s − 1.26·10-s + 1/4·16-s − 0.458·19-s + 0.894·20-s − 0.417·23-s + 11/5·25-s + 0.742·29-s − 0.176·32-s + 0.324·38-s − 0.632·40-s + 0.609·43-s + 0.294·46-s + 1.75·47-s + 4/7·49-s − 1.55·50-s + 2.19·53-s − 0.525·58-s + 1/8·64-s − 1.22·67-s − 0.237·71-s − 0.468·73-s − 0.229·76-s + 0.447·80-s + ⋯

Functional equation

Λ(s)=(259200s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(259200s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 259200259200    =    27⋅34⋅522^{7} \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 16.526816.5268
Root analytic conductor: 2.016262.01626
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 259200, ( :1/2,1/2), 1)(4,\ 259200,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8421176821.842117682
L(12)L(\frac12) ≈\approx 1.8421176821.842117682
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
3 1 1
5C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good7C22C_2^2 1−4T2+p2T4 1 - 4 T^{2} + p^{2} T^{4}
11C22C_2^2 1−20T2+p2T4 1 - 20 T^{2} + p^{2} T^{4}
13C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
17C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1−4T+pT2)(1+6T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
23C2C_2×\timesC2C_2 (1−4T+pT2)(1+6T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2×\timesC2C_2 (1−6T+pT2)(1+2T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
37C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
41C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2×\timesC2C_2 (1−8T+pT2)(1−4T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} )
53C2C_2×\timesC2C_2 (1−10T+pT2)(1−6T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} )
59C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
61C22C_2^2 1−38T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1+pT2)(1+10T+pT2) ( 1 + p T^{2} )( 1 + 10 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−6T+pT2)(1+8T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C22C_2^2 1−142T2+p2T4 1 - 142 T^{2} + p^{2} T^{4}
83C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
89C22C_2^2 1−74T2+p2T4 1 - 74 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−2T+pT2)(1+12T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 12 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

−9.081444259937376947085319925841, −8.523285408969769437469971320630, −8.228116244396256443869053669012, −7.34470332136653973429739824837, −7.11957178074071411009733756910, −6.54351622473128791008628510863, −5.94692085854568052038423647429, −5.77270774455158360582234041973, −5.22074566980770104380699023481, −4.46117196245196832818359739611, −3.88296031384893801501633766656, −2.81751947676589648650979183563, −2.48071790774233938628729689497, −1.79620548558312040514035893609, −0.961937939221127316723838522284, 0.961937939221127316723838522284, 1.79620548558312040514035893609, 2.48071790774233938628729689497, 2.81751947676589648650979183563, 3.88296031384893801501633766656, 4.46117196245196832818359739611, 5.22074566980770104380699023481, 5.77270774455158360582234041973, 5.94692085854568052038423647429, 6.54351622473128791008628510863, 7.11957178074071411009733756910, 7.34470332136653973429739824837, 8.228116244396256443869053669012, 8.523285408969769437469971320630, 9.081444259937376947085319925841

Graph of the ZZ-function along the critical line