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Properties

Label 4-243675-1.1-c1e2-0-2
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  − 2·2-s − 3-s − 4-s + 2·6-s + 8·8-s + 9-s + 12-s − 7·16-s − 2·18-s + 4·19-s − 8·24-s + 25-s − 27-s − 4·29-s − 14·32-s − 36-s − 8·38-s + 20·41-s + 8·43-s + 7·48-s − 14·49-s − 2·50-s − 20·53-s + 2·54-s − 4·57-s + 8·58-s − 8·59-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s − 1/2·4-s + 0.816·6-s + 2.82·8-s + 1/3·9-s + 0.288·12-s − 7/4·16-s − 0.471·18-s + 0.917·19-s − 1.63·24-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 2.47·32-s − 1/6·36-s − 1.29·38-s + 3.12·41-s + 1.21·43-s + 1.01·48-s − 2·49-s − 0.282·50-s − 2.74·53-s + 0.272·54-s − 0.529·57-s + 1.05·58-s − 1.04·59-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 )
19C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good2C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
7C2C_2 (1+pT2)2 ( 1 + 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 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 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+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
59C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{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+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{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

−8.896447809747983381121393959041, −8.232011963291614168935511350860, −7.77071280677534797445852255244, −7.66488013441745380243230842523, −7.11524013136289051004798684067, −6.34145167556094658057083324143, −5.83679390833615760339265246745, −5.23920392624592057055772361749, −4.72109247762374245491183934060, −4.33195666036117235086666220277, −3.72492686467354363460541892516, −2.85380672919944569836723726469, −1.66341921070628603647910394136, −1.05029361783004892799606717843, 0, 1.05029361783004892799606717843, 1.66341921070628603647910394136, 2.85380672919944569836723726469, 3.72492686467354363460541892516, 4.33195666036117235086666220277, 4.72109247762374245491183934060, 5.23920392624592057055772361749, 5.83679390833615760339265246745, 6.34145167556094658057083324143, 7.11524013136289051004798684067, 7.66488013441745380243230842523, 7.77071280677534797445852255244, 8.232011963291614168935511350860, 8.896447809747983381121393959041

Graph of the ZZ-function along the critical line