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Properties

Label 4-225792-1.1-c1e2-0-29
Degree 44
Conductor 225792225792
Sign −1-1
Analytic cond. 14.396614.3966
Root an. cond. 1.947891.94789
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 3·9-s + 16·19-s + 2·25-s − 12·29-s − 8·43-s + 12·45-s + 16·47-s + 49-s − 12·53-s − 8·67-s + 16·71-s + 20·73-s + 9·81-s − 64·95-s − 12·97-s − 4·101-s − 6·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1.78·5-s − 9-s + 3.67·19-s + 2/5·25-s − 2.22·29-s − 1.21·43-s + 1.78·45-s + 2.33·47-s + 1/7·49-s − 1.64·53-s − 0.977·67-s + 1.89·71-s + 2.34·73-s + 81-s − 6.56·95-s − 1.21·97-s − 0.398·101-s − 0.545·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

Λ(s)=(225792s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225792 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(225792s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225792 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 225792225792    =    29⋅32⋅722^{9} \cdot 3^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 14.396614.3966
Root analytic conductor: 1.947891.94789
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 225792, ( :1/2,1/2), −1)(4,\ 225792,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
3C2C_2 1+pT2 1 + p T^{2}
7C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good5C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
83C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1+6T+pT2)2 ( 1 + 6 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

−8.749304120934004278681759852831, −8.093537689649372229665039418958, −7.79292083571821938290976650763, −7.40397429472305265033368100557, −7.26382021666693896035142333427, −6.35373904523042773985731435784, −5.69394971603405508863657314834, −5.29340219049011126125230323142, −4.92225264716551837532790929786, −3.89550776474805329328185747620, −3.60173413237973468362473708205, −3.26735667825841356907489548421, −2.40145156924384381758797773878, −1.15037892508990574535308118020, 0, 1.15037892508990574535308118020, 2.40145156924384381758797773878, 3.26735667825841356907489548421, 3.60173413237973468362473708205, 3.89550776474805329328185747620, 4.92225264716551837532790929786, 5.29340219049011126125230323142, 5.69394971603405508863657314834, 6.35373904523042773985731435784, 7.26382021666693896035142333427, 7.40397429472305265033368100557, 7.79292083571821938290976650763, 8.093537689649372229665039418958, 8.749304120934004278681759852831

Graph of the ZZ-function along the critical line