↑

Properties

Label 4-288000-1.1-c1e2-0-21
Degree 44
Conductor 288000288000
Sign −1-1
Analytic cond. 18.363118.3631
Root an. cond. 2.070072.07007
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 9-s + 25-s − 12·29-s − 20·41-s − 45-s − 2·49-s − 12·61-s + 81-s + 4·89-s − 28·101-s + 20·109-s − 22·121-s + 125-s + 127-s + 131-s + 137-s + 139-s − 12·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 0.447·5-s − 1/3·9-s + 1/5·25-s − 2.22·29-s − 3.12·41-s − 0.149·45-s − 2/7·49-s − 1.53·61-s + 1/9·81-s + 0.423·89-s − 2.78·101-s + 1.91·109-s − 2·121-s + 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.996·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

Λ(s)=(288000s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(288000s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 288000288000    =    28⋅32⋅532^{8} \cdot 3^{2} \cdot 5^{3}
Sign: −1-1
Analytic conductor: 18.363118.3631
Root analytic conductor: 2.070072.07007
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 288000, ( :1/2,1/2), −1)(4,\ 288000,\ (\ :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)
bad2 1 1
3C2C_2 1+T2 1 + T^{2}
5C1C_1 1−T 1 - T
good7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
29C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
67C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
79C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
83C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
89C2C_2 (1−2T+pT2)2 ( 1 - 2 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} )
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

−8.651398621157294016754784166904, −8.177011345230851378990633313291, −7.71733313943630569596593201575, −7.20437669814029042430609930443, −6.68180261171334745171727340894, −6.30690797191976477769025407425, −5.57771460248008481046830603443, −5.37353169568061066443481917418, −4.77118490049532217723449569095, −4.07093718664689551191991169282, −3.44839557200100878301137886825, −2.97418651820743721837781579798, −2.04012192925769453409315274550, −1.53830675039595379013924667510, 0, 1.53830675039595379013924667510, 2.04012192925769453409315274550, 2.97418651820743721837781579798, 3.44839557200100878301137886825, 4.07093718664689551191991169282, 4.77118490049532217723449569095, 5.37353169568061066443481917418, 5.57771460248008481046830603443, 6.30690797191976477769025407425, 6.68180261171334745171727340894, 7.20437669814029042430609930443, 7.71733313943630569596593201575, 8.177011345230851378990633313291, 8.651398621157294016754784166904

Graph of the ZZ-function along the critical line