↑

Properties

Label 4-209952-1.1-c1e2-0-20
Degree 44
Conductor 209952209952
Sign −1-1
Analytic cond. 13.386713.3867
Root an. cond. 1.912791.91279
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 6·11-s + 4·13-s + 16-s − 6·22-s − 12·23-s − 10·25-s + 4·26-s + 32-s − 8·37-s − 6·44-s − 12·46-s − 12·47-s − 10·49-s − 10·50-s + 4·52-s + 6·59-s + 16·61-s + 64-s − 24·71-s + 22·73-s − 8·74-s + 24·83-s − 6·88-s − 12·92-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.80·11-s + 1.10·13-s + 1/4·16-s − 1.27·22-s − 2.50·23-s − 2·25-s + 0.784·26-s + 0.176·32-s − 1.31·37-s − 0.904·44-s − 1.76·46-s − 1.75·47-s − 1.42·49-s − 1.41·50-s + 0.554·52-s + 0.781·59-s + 2.04·61-s + 1/8·64-s − 2.84·71-s + 2.57·73-s − 0.929·74-s + 2.63·83-s − 0.639·88-s − 1.25·92-s + ⋯

Functional equation

Λ(s)=(209952s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(209952s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 209952209952    =    25⋅382^{5} \cdot 3^{8}
Sign: −1-1
Analytic conductor: 13.386713.3867
Root analytic conductor: 1.912791.91279
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 209952, ( :1/2,1/2), −1)(4,\ 209952,\ (\ :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)
bad2C1C_1 1−T 1 - T
3 1 1
good5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
19C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
23C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
41C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
43C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
47C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
53C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
59C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
61C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
67C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
71C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
73C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
79C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C2C_2 (1−12T+pT2)2 ( 1 - 12 T + 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 (1−5T+pT2)2 ( 1 - 5 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

−8.543319215757720981308384286559, −8.255589340941165804909076040404, −7.79453288119723697641112445352, −7.63671497466111343831458484773, −6.69953527334383655382904548898, −6.25936593777468405983226275094, −5.91234365712937538559209143490, −5.13805326110939411578475614907, −5.13777464868847606985261506723, −3.94847330019587176084970982031, −3.86098888246252906299834258338, −3.10746305536689930220572132603, −2.19879385611801234222671009191, −1.80773362483917328069437252787, 0, 1.80773362483917328069437252787, 2.19879385611801234222671009191, 3.10746305536689930220572132603, 3.86098888246252906299834258338, 3.94847330019587176084970982031, 5.13777464868847606985261506723, 5.13805326110939411578475614907, 5.91234365712937538559209143490, 6.25936593777468405983226275094, 6.69953527334383655382904548898, 7.63671497466111343831458484773, 7.79453288119723697641112445352, 8.255589340941165804909076040404, 8.543319215757720981308384286559

Graph of the ZZ-function along the critical line