↑

Properties

Label 4-960e2-1.1-c1e2-0-36
Degree 44
Conductor 921600921600
Sign 11
Analytic cond. 58.762058.7620
Root an. cond. 2.768682.76868
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  + 4·5-s − 9-s + 4·11-s + 11·25-s + 16·31-s + 4·41-s − 4·45-s + 10·49-s + 16·55-s − 20·59-s − 4·61-s − 24·71-s + 81-s + 20·89-s − 4·99-s + 16·101-s + 20·109-s − 10·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 64·155-s + 157-s + ⋯
L(s)  = 1  + 1.78·5-s − 1/3·9-s + 1.20·11-s + 11/5·25-s + 2.87·31-s + 0.624·41-s − 0.596·45-s + 10/7·49-s + 2.15·55-s − 2.60·59-s − 0.512·61-s − 2.84·71-s + 1/9·81-s + 2.11·89-s − 0.402·99-s + 1.59·101-s + 1.91·109-s − 0.909·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 5.14·155-s + 0.0798·157-s + ⋯

Functional equation

Λ(s)=(921600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(921600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 921600921600    =    212⋅32⋅522^{12} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 58.762058.7620
Root analytic conductor: 2.768682.76868
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 921600, ( :1/2,1/2), 1)(4,\ 921600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.5215630733.521563073
L(12)L(\frac12) ≈\approx 3.5215630733.521563073
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}
5C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1−2T+pT2)2 ( 1 - 2 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}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
71C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
73C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
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}
97C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 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

−10.18364502159992335238648205135, −9.833776562646347466676882747811, −9.384064569249551478990180713425, −8.971978897568705580423844699306, −8.826894411114671847087748345650, −8.322158042952626455600158208771, −7.55950415037675494715112703776, −7.37690217813721676929007350833, −6.54196454830521372494982997938, −6.26610215086726361520580870028, −6.12787929568860191645301468827, −5.66263154031096757822641733587, −4.87625642194862086874035110455, −4.67011774542556332856712106536, −4.08235065221894392875316020330, −3.24016238096554676569682538509, −2.79037509052512667748577503659, −2.25673511107964657709754938345, −1.50590960483716422198531697519, −0.975660376270329757854206180532, 0.975660376270329757854206180532, 1.50590960483716422198531697519, 2.25673511107964657709754938345, 2.79037509052512667748577503659, 3.24016238096554676569682538509, 4.08235065221894392875316020330, 4.67011774542556332856712106536, 4.87625642194862086874035110455, 5.66263154031096757822641733587, 6.12787929568860191645301468827, 6.26610215086726361520580870028, 6.54196454830521372494982997938, 7.37690217813721676929007350833, 7.55950415037675494715112703776, 8.322158042952626455600158208771, 8.826894411114671847087748345650, 8.971978897568705580423844699306, 9.384064569249551478990180713425, 9.833776562646347466676882747811, 10.18364502159992335238648205135

Graph of the ZZ-function along the critical line