↑

Properties

Label 4-4608e2-1.1-c1e2-0-47
Degree 44
Conductor 2123366421233664
Sign 11
Analytic cond. 1353.871353.87
Root an. cond. 6.065896.06589
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  − 8·11-s − 12·13-s + 12·23-s + 10·25-s − 12·37-s − 12·47-s − 4·49-s + 8·59-s + 12·61-s − 12·71-s + 12·73-s − 32·83-s − 24·97-s − 32·107-s + 12·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 96·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + ⋯
L(s)  = 1  − 2.41·11-s − 3.32·13-s + 2.50·23-s + 2·25-s − 1.97·37-s − 1.75·47-s − 4/7·49-s + 1.04·59-s + 1.53·61-s − 1.42·71-s + 1.40·73-s − 3.51·83-s − 2.43·97-s − 3.09·107-s + 1.14·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8.02·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + ⋯

Functional equation

Λ(s)=(21233664s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(21233664s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2123366421233664    =    218⋅342^{18} \cdot 3^{4}
Sign: 11
Analytic conductor: 1353.871353.87
Root analytic conductor: 6.065896.06589
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 21233664, ( :1/2,1/2), 1)(4,\ 21233664,\ (\ :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
good5C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
7C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
11C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
13C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
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 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
29C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
31C22C_2^2 1−44T2+p2T4 1 - 44 T^{2} + p^{2} T^{4}
37C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
41C22C_2^2 1−80T2+p2T4 1 - 80 T^{2} + p^{2} T^{4}
43C22C_2^2 1−78T2+p2T4 1 - 78 T^{2} + p^{2} T^{4}
47C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
53C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
59C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
61C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
67C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
71C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
79C22C_2^2 1−140T2+p2T4 1 - 140 T^{2} + p^{2} T^{4}
83C2C_2 (1+16T+pT2)2 ( 1 + 16 T + p T^{2} )^{2}
89C22C_2^2 1−16T2+p2T4 1 - 16 T^{2} + p^{2} T^{4}
97C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{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

−8.184119526784080096930054583337, −7.76714131083724239901865740792, −7.28663558619794145335692789488, −7.02491795510718615019431936981, −6.83405135447474564037169283022, −6.72830861693029482285171102857, −5.59738193717085894569396855219, −5.30809244850357323314924184756, −5.21799090855866861698966030986, −4.96981545298691664253380339263, −4.61752563126245202166051772557, −4.16502054511091565788199569360, −3.11753878681541518270339046385, −3.08596394900680291147734248050, −2.68276431274612271923823533354, −2.43384670690819110329111151351, −1.78827932214867660260249596332, −1.01465676791758919550831411091, 0, 0, 1.01465676791758919550831411091, 1.78827932214867660260249596332, 2.43384670690819110329111151351, 2.68276431274612271923823533354, 3.08596394900680291147734248050, 3.11753878681541518270339046385, 4.16502054511091565788199569360, 4.61752563126245202166051772557, 4.96981545298691664253380339263, 5.21799090855866861698966030986, 5.30809244850357323314924184756, 5.59738193717085894569396855219, 6.72830861693029482285171102857, 6.83405135447474564037169283022, 7.02491795510718615019431936981, 7.28663558619794145335692789488, 7.76714131083724239901865740792, 8.184119526784080096930054583337

Graph of the ZZ-function along the critical line