↑

Properties

Label 4-9702e2-1.1-c1e2-0-23
Degree 44
Conductor 9412880494128804
Sign 11
Analytic cond. 6001.736001.73
Root an. cond. 8.801758.80175
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  − 2·2-s + 3·4-s − 4·8-s − 2·11-s + 5·16-s + 4·22-s + 8·23-s − 8·25-s + 12·29-s − 6·32-s − 16·37-s − 16·43-s − 6·44-s − 16·46-s + 16·50-s − 24·58-s + 7·64-s − 16·67-s + 16·71-s + 32·74-s + 8·79-s + 32·86-s + 8·88-s + 24·92-s − 24·100-s − 16·107-s + 8·109-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s − 0.603·11-s + 5/4·16-s + 0.852·22-s + 1.66·23-s − 8/5·25-s + 2.22·29-s − 1.06·32-s − 2.63·37-s − 2.43·43-s − 0.904·44-s − 2.35·46-s + 2.26·50-s − 3.15·58-s + 7/8·64-s − 1.95·67-s + 1.89·71-s + 3.71·74-s + 0.900·79-s + 3.45·86-s + 0.852·88-s + 2.50·92-s − 2.39·100-s − 1.54·107-s + 0.766·109-s + ⋯

Functional equation

Λ(s)=(94128804s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(94128804s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9412880494128804    =    22⋅34⋅74⋅1122^{2} \cdot 3^{4} \cdot 7^{4} \cdot 11^{2}
Sign: 11
Analytic conductor: 6001.736001.73
Root analytic conductor: 8.801758.80175
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 94128804, ( :1/2,1/2), 1)(4,\ 94128804,\ (\ :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)
bad2C1C_1 (1+T)2 ( 1 + T )^{2}
3 1 1
7 1 1
11C1C_1 (1+T)2 ( 1 + T )^{2}
good5C22C_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+32T2+p2T4 1 + 32 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 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C22C_2^2 1+54T2+p2T4 1 + 54 T^{2} + p^{2} T^{4}
37C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
41C22C_2^2 1+32T2+p2T4 1 + 32 T^{2} + p^{2} T^{4}
43C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
47C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C22C_2^2 1+72T2+p2T4 1 + 72 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 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
83C22C_2^2 1+158T2+p2T4 1 + 158 T^{2} + p^{2} T^{4}
89C22C_2^2 1+80T2+p2T4 1 + 80 T^{2} + p^{2} T^{4}
97C22C_2^2 1−48T2+p2T4 1 - 48 T^{2} + 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

−7.44101026525792074041023290162, −7.28221579696518488307160152922, −6.83241933588875437708540172818, −6.76647536949665559531313365175, −6.15855033748627495505040230747, −6.11686917129522057351557448824, −5.35074155741522510869068543679, −5.21118544326895154803339956925, −4.85308670455194117520888050716, −4.52228542407737730630259468249, −3.69952845422182766797975115257, −3.58566315054123885461219146000, −2.99428346365492788390692356300, −2.86985277158359714171875405724, −2.06572600660909889910839720824, −2.04319898960403537319275019497, −1.23854838287504997368745280975, −1.06399021800454625787883543368, 0, 0, 1.06399021800454625787883543368, 1.23854838287504997368745280975, 2.04319898960403537319275019497, 2.06572600660909889910839720824, 2.86985277158359714171875405724, 2.99428346365492788390692356300, 3.58566315054123885461219146000, 3.69952845422182766797975115257, 4.52228542407737730630259468249, 4.85308670455194117520888050716, 5.21118544326895154803339956925, 5.35074155741522510869068543679, 6.11686917129522057351557448824, 6.15855033748627495505040230747, 6.76647536949665559531313365175, 6.83241933588875437708540172818, 7.28221579696518488307160152922, 7.44101026525792074041023290162

Graph of the ZZ-function along the critical line