↑

Properties

Label 4-243675-1.1-c1e2-0-1
Degree 44
Conductor 243675243675
Sign 11
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 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·4-s + 8·7-s + 9-s + 3·12-s + 4·13-s + 5·16-s − 2·19-s − 8·21-s + 25-s − 27-s − 24·28-s − 3·36-s − 12·37-s − 4·39-s + 16·43-s − 5·48-s + 34·49-s − 12·52-s + 2·57-s + 28·61-s + 8·63-s − 3·64-s − 8·67-s − 28·73-s − 75-s + 6·76-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s + 3.02·7-s + 1/3·9-s + 0.866·12-s + 1.10·13-s + 5/4·16-s − 0.458·19-s − 1.74·21-s + 1/5·25-s − 0.192·27-s − 4.53·28-s − 1/2·36-s − 1.97·37-s − 0.640·39-s + 2.43·43-s − 0.721·48-s + 34/7·49-s − 1.66·52-s + 0.264·57-s + 3.58·61-s + 1.00·63-s − 3/8·64-s − 0.977·67-s − 3.27·73-s − 0.115·75-s + 0.688·76-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: 11
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: 00
Selberg data: (4, 243675, ( :1/2,1/2), 1)(4,\ 243675,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5331474751.533147475
L(12)L(\frac12) ≈\approx 1.5331474751.533147475
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 (1−4T+pT2)2 ( 1 - 4 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)2 ( 1 - 2 T + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
47C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
53C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
79C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{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

−8.956131697052494305782515642843, −8.576861436588155713465515075006, −8.056447043933612031171269197880, −7.79111029211693930669614606595, −7.22090991652715915706448156971, −6.54655367252788500605108631348, −5.59885321084019164088416378698, −5.57442263951836682011583202205, −4.80915338219819536204131275847, −4.73512523477553586192350141458, −4.00775128266596845974693279239, −3.75150945152824378664368493386, −2.32830304228468473527730093464, −1.55903104009719190466896724014, −0.908717833826787650120905118138, 0.908717833826787650120905118138, 1.55903104009719190466896724014, 2.32830304228468473527730093464, 3.75150945152824378664368493386, 4.00775128266596845974693279239, 4.73512523477553586192350141458, 4.80915338219819536204131275847, 5.57442263951836682011583202205, 5.59885321084019164088416378698, 6.54655367252788500605108631348, 7.22090991652715915706448156971, 7.79111029211693930669614606595, 8.056447043933612031171269197880, 8.576861436588155713465515075006, 8.956131697052494305782515642843

Graph of the ZZ-function along the critical line