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Properties

Label 4-243675-1.1-c1e2-0-3
Degree 44
Conductor 243675243675
Sign −1-1
Analytic cond. 15.536915.5369
Root an. cond. 1.985361.98536
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  + 3-s − 3·4-s − 4·7-s + 9-s − 3·12-s + 5·16-s + 2·19-s − 4·21-s + 25-s + 27-s + 12·28-s − 3·36-s + 8·37-s − 4·43-s + 5·48-s − 2·49-s + 2·57-s + 4·61-s − 4·63-s − 3·64-s − 16·67-s + 28·73-s + 75-s − 6·76-s + 16·79-s + 81-s + 12·84-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s − 1.51·7-s + 1/3·9-s − 0.866·12-s + 5/4·16-s + 0.458·19-s − 0.872·21-s + 1/5·25-s + 0.192·27-s + 2.26·28-s − 1/2·36-s + 1.31·37-s − 0.609·43-s + 0.721·48-s − 2/7·49-s + 0.264·57-s + 0.512·61-s − 0.503·63-s − 3/8·64-s − 1.95·67-s + 3.27·73-s + 0.115·75-s − 0.688·76-s + 1.80·79-s + 1/9·81-s + 1.30·84-s + ⋯

Functional equation

Λ(s)=(243675s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(243675s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 243675243675    =    33⋅52⋅1923^{3} \cdot 5^{2} \cdot 19^{2}
Sign: −1-1
Analytic conductor: 15.536915.5369
Root analytic conductor: 1.985361.98536
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 243675, ( :1/2,1/2), −1)(4,\ 243675,\ (\ :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)
bad3C1C_1 1−T 1 - T
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
19C1C_1 (1−T)2 ( 1 - T )^{2}
good2C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
7C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
11C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
71C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
73C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
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

−8.836488963226532523388598253810, −8.311172749518175221259006108214, −7.944956574862892187546933534941, −7.44218328127442158412477604159, −6.69343741358824607018216997419, −6.43277558403273238107542389585, −5.81033642723526554045616411588, −5.08269165937541766478904797647, −4.81376783679940109514525317705, −3.93065235513561826916389859535, −3.73246630802271223857044732290, −3.06388291591923378327344798065, −2.45875404104195454497297562929, −1.15112547813740378682858966686, 0, 1.15112547813740378682858966686, 2.45875404104195454497297562929, 3.06388291591923378327344798065, 3.73246630802271223857044732290, 3.93065235513561826916389859535, 4.81376783679940109514525317705, 5.08269165937541766478904797647, 5.81033642723526554045616411588, 6.43277558403273238107542389585, 6.69343741358824607018216997419, 7.44218328127442158412477604159, 7.944956574862892187546933534941, 8.311172749518175221259006108214, 8.836488963226532523388598253810

Graph of the ZZ-function along the critical line