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Properties

Label 4-8624e2-1.1-c1e2-0-24
Degree 44
Conductor 7437337674373376
Sign 11
Analytic cond. 4742.114742.11
Root an. cond. 8.298378.29837
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s + 2·11-s − 8·23-s − 8·25-s − 16·37-s − 8·43-s − 12·53-s + 16·67-s − 16·71-s − 32·79-s − 5·81-s + 4·99-s + 16·107-s + 32·109-s + 12·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + ⋯
L(s)  = 1  + 2/3·9-s + 0.603·11-s − 1.66·23-s − 8/5·25-s − 2.63·37-s − 1.21·43-s − 1.64·53-s + 1.95·67-s − 1.89·71-s − 3.60·79-s − 5/9·81-s + 0.402·99-s + 1.54·107-s + 3.06·109-s + 1.12·113-s + 3/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-s + ⋯

Functional equation

Λ(s)=(74373376s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(74373376s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 7437337674373376    =    28⋅74⋅1122^{8} \cdot 7^{4} \cdot 11^{2}
Sign: 11
Analytic conductor: 4742.114742.11
Root analytic conductor: 8.298378.29837
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 74373376, ( :1/2,1/2), 1)(4,\ 74373376,\ (\ :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
7 1 1
11C1C_1 (1−T)2 ( 1 - T )^{2}
good3C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
5C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + 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}
19C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
23C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
37C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
41C22C_2^2 1−16T2+p2T4 1 - 16 T^{2} + p^{2} T^{4}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
61C22C_2^2 1+120T2+p2T4 1 + 120 T^{2} + p^{2} T^{4}
67C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
71C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
73C22C_2^2 1+144T2+p2T4 1 + 144 T^{2} + p^{2} T^{4}
79C2C_2 (1+16T+pT2)2 ( 1 + 16 T + p T^{2} )^{2}
83C22C_2^2 1+158T2+p2T4 1 + 158 T^{2} + p^{2} T^{4}
89C22C_2^2 1−64T2+p2T4 1 - 64 T^{2} + p^{2} T^{4}
97C22C_2^2 1+96T2+p2T4 1 + 96 T^{2} + p^{2} T^{4}
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   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

−7.46760761991361247988733304208, −7.29539316156711661568897171555, −7.04413911466501462583926811631, −6.47726598872353544502337158923, −6.26009793297953143686702153148, −5.95446841228358925288734060977, −5.56911407749023556267444126448, −5.17855580284254959492776254623, −4.79348895076068348219361886611, −4.29657086124467134879326308352, −4.13480041904322698132155791797, −3.70980917516283042942945851155, −3.23965352148593878850875245520, −3.10662169369221883707936802326, −2.18229187698172178580742072031, −1.91441368626081719640283111893, −1.64898519759039438656792173194, −1.12113155743030734812238244327, 0, 0, 1.12113155743030734812238244327, 1.64898519759039438656792173194, 1.91441368626081719640283111893, 2.18229187698172178580742072031, 3.10662169369221883707936802326, 3.23965352148593878850875245520, 3.70980917516283042942945851155, 4.13480041904322698132155791797, 4.29657086124467134879326308352, 4.79348895076068348219361886611, 5.17855580284254959492776254623, 5.56911407749023556267444126448, 5.95446841228358925288734060977, 6.26009793297953143686702153148, 6.47726598872353544502337158923, 7.04413911466501462583926811631, 7.29539316156711661568897171555, 7.46760761991361247988733304208

Graph of the ZZ-function along the critical line