↑

Properties

Label 4-240e2-1.1-c1e2-0-6
Degree 44
Conductor 5760057600
Sign 11
Analytic cond. 3.672623.67262
Root an. cond. 1.384341.38434
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  − 2·2-s + 2·4-s − 4·5-s + 6·7-s − 9-s + 8·10-s + 2·11-s + 8·13-s − 12·14-s − 4·16-s + 6·17-s + 2·18-s − 6·19-s − 8·20-s − 4·22-s − 2·23-s + 11·25-s − 16·26-s + 12·28-s + 6·29-s + 8·32-s − 12·34-s − 24·35-s − 2·36-s + 16·37-s + 12·38-s − 12·43-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.78·5-s + 2.26·7-s − 1/3·9-s + 2.52·10-s + 0.603·11-s + 2.21·13-s − 3.20·14-s − 16-s + 1.45·17-s + 0.471·18-s − 1.37·19-s − 1.78·20-s − 0.852·22-s − 0.417·23-s + 11/5·25-s − 3.13·26-s + 2.26·28-s + 1.11·29-s + 1.41·32-s − 2.05·34-s − 4.05·35-s − 1/3·36-s + 2.63·37-s + 1.94·38-s − 1.82·43-s + ⋯

Functional equation

Λ(s)=(57600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(57600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5760057600    =    28⋅32⋅522^{8} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 3.672623.67262
Root analytic conductor: 1.384341.38434
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 57600, ( :1/2,1/2), 1)(4,\ 57600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.73937761480.7393776148
L(12)L(\frac12) ≈\approx 0.73937761480.7393776148
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)
bad2C2C_2 1+pT+pT2 1 + p T + p T^{2}
3C2C_2 1+T2 1 + T^{2}
5C2C_2 1+4T+pT2 1 + 4 T + p T^{2}
good7C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
17C2C_2 (1−8T+pT2)(1+2T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
29C2C_2 (1−10T+pT2)(1+4T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} )
31C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
37C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
41C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
43C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
47C22C_2^2 1−10T+50T2−10pT3+p2T4 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C22C_2^2 1+14T+98T2+14pT3+p2T4 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+18T+162T2+18pT3+p2T4 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4}
67C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
71C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)(1+16T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1−150T2+p2T4 1 - 150 T^{2} + p^{2} T^{4}
89C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
97C22C_2^2 1+14T+98T2+14pT3+p2T4 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4}
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

−11.96024738045319848409027760090, −11.67709057056776640608665440554, −11.25747202673573245353186410792, −11.05892076509857134572949158698, −10.48926098466320785324298423132, −10.24645145614500882936019503435, −9.022944640304182734043171508817, −8.847102673013975322189811811695, −8.365898651857278957392919164717, −8.106713653208418927705469847711, −7.58987157855732880100126859742, −7.50217387271728220135325116551, −6.23036938681267556806264616869, −6.12505135261452938585231677078, −4.78240645545692724880683043666, −4.46951425297280730473998307454, −3.94419874086849339702377654713, −3.03414532889574315065370727459, −1.59657543727113414694446654062, −1.05186215761897664409518131513, 1.05186215761897664409518131513, 1.59657543727113414694446654062, 3.03414532889574315065370727459, 3.94419874086849339702377654713, 4.46951425297280730473998307454, 4.78240645545692724880683043666, 6.12505135261452938585231677078, 6.23036938681267556806264616869, 7.50217387271728220135325116551, 7.58987157855732880100126859742, 8.106713653208418927705469847711, 8.365898651857278957392919164717, 8.847102673013975322189811811695, 9.022944640304182734043171508817, 10.24645145614500882936019503435, 10.48926098466320785324298423132, 11.05892076509857134572949158698, 11.25747202673573245353186410792, 11.67709057056776640608665440554, 11.96024738045319848409027760090

Graph of the ZZ-function along the critical line