↑

Properties

Label 4-3960e2-1.1-c1e2-0-13
Degree 44
Conductor 1568160015681600
Sign 11
Analytic cond. 999.872999.872
Root an. cond. 5.623235.62323
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 4·7-s + 2·11-s − 8·19-s + 3·25-s − 4·29-s − 4·31-s − 8·35-s + 4·37-s − 4·41-s − 4·43-s − 8·47-s − 4·53-s + 4·55-s − 12·59-s + 12·61-s − 4·71-s − 8·77-s − 16·79-s − 12·83-s − 4·89-s − 16·95-s + 12·97-s − 20·101-s − 12·107-s − 4·109-s − 4·113-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.51·7-s + 0.603·11-s − 1.83·19-s + 3/5·25-s − 0.742·29-s − 0.718·31-s − 1.35·35-s + 0.657·37-s − 0.624·41-s − 0.609·43-s − 1.16·47-s − 0.549·53-s + 0.539·55-s − 1.56·59-s + 1.53·61-s − 0.474·71-s − 0.911·77-s − 1.80·79-s − 1.31·83-s − 0.423·89-s − 1.64·95-s + 1.21·97-s − 1.99·101-s − 1.16·107-s − 0.383·109-s − 0.376·113-s + ⋯

Functional equation

Λ(s)=(15681600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(15681600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1568160015681600    =    26⋅34⋅52⋅1122^{6} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 999.872999.872
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 15681600, ( :1/2,1/2), 1)(4,\ 15681600,\ (\ :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
3 1 1
5C1C_1 (1−T)2 ( 1 - T )^{2}
11C1C_1 (1−T)2 ( 1 - T )^{2}
good7D4D_{4} 1+4T+16T2+4pT3+p2T4 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4}
13C22C_2^2 1+24T2+p2T4 1 + 24 T^{2} + p^{2} T^{4}
17C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
19D4D_{4} 1+8T+46T2+8pT3+p2T4 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
29D4D_{4} 1+4T+30T2+4pT3+p2T4 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+4T+58T2+4pT3+p2T4 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−4T+46T2−4pT3+p2T4 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+4T+54T2+4pT3+p2T4 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+4T+40T2+4pT3+p2T4 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+8T+38T2+8pT3+p2T4 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+4T+38T2+4pT3+p2T4 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+12T+146T2+12pT3+p2T4 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−12T+150T2−12pT3+p2T4 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
71D4D_{4} 1+4T+74T2+4pT3+p2T4 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+96T2+p2T4 1 + 96 T^{2} + p^{2} T^{4}
79D4D_{4} 1+16T+150T2+16pT3+p2T4 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+12T+184T2+12pT3+p2T4 1 + 12 T + 184 T^{2} + 12 p T^{3} + p^{2} T^{4}
89C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
97D4D_{4} 1−12T+222T2−12pT3+p2T4 1 - 12 T + 222 T^{2} - 12 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

−8.194355030886597087753601489674, −8.163684674078997994178342669140, −7.24642568476115040161137961396, −7.24061560018978877647641821837, −6.61677892911359614889753164693, −6.40803401000670299423274467730, −6.16696095168987411800175414965, −5.90088373427917014200973283116, −5.18830216875547986691653458756, −5.10186981458566473880135099115, −4.26631656801897963867301040378, −4.17669831550017246343552535878, −3.51168180516876892914451575964, −3.26746097291734380070123010035, −2.64410486088545206910864162385, −2.35306552808578843653954551269, −1.56632787403980689866663588515, −1.41389604083740205549666632957, 0, 0, 1.41389604083740205549666632957, 1.56632787403980689866663588515, 2.35306552808578843653954551269, 2.64410486088545206910864162385, 3.26746097291734380070123010035, 3.51168180516876892914451575964, 4.17669831550017246343552535878, 4.26631656801897963867301040378, 5.10186981458566473880135099115, 5.18830216875547986691653458756, 5.90088373427917014200973283116, 6.16696095168987411800175414965, 6.40803401000670299423274467730, 6.61677892911359614889753164693, 7.24061560018978877647641821837, 7.24642568476115040161137961396, 8.163684674078997994178342669140, 8.194355030886597087753601489674

Graph of the ZZ-function along the critical line