↑

Properties

Label 4-7488e2-1.1-c1e2-0-15
Degree 44
Conductor 5607014456070144
Sign 11
Analytic cond. 3575.083575.08
Root an. cond. 7.732527.73252
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  − 3·5-s + 3·7-s + 4·11-s + 2·13-s − 3·17-s − 12·19-s + 25-s + 6·31-s − 9·35-s + 7·37-s − 6·41-s − 3·43-s + 9·47-s − 3·49-s − 18·53-s − 12·55-s − 12·59-s − 2·61-s − 6·65-s − 12·67-s − 9·71-s + 20·73-s + 12·77-s + 24·79-s − 10·83-s + 9·85-s + 6·91-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.13·7-s + 1.20·11-s + 0.554·13-s − 0.727·17-s − 2.75·19-s + 1/5·25-s + 1.07·31-s − 1.52·35-s + 1.15·37-s − 0.937·41-s − 0.457·43-s + 1.31·47-s − 3/7·49-s − 2.47·53-s − 1.61·55-s − 1.56·59-s − 0.256·61-s − 0.744·65-s − 1.46·67-s − 1.06·71-s + 2.34·73-s + 1.36·77-s + 2.70·79-s − 1.09·83-s + 0.976·85-s + 0.628·91-s + ⋯

Functional equation

Λ(s)=(56070144s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(56070144s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5607014456070144    =    212⋅34⋅1322^{12} \cdot 3^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 3575.083575.08
Root analytic conductor: 7.732527.73252
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 56070144, ( :1/2,1/2), 1)(4,\ 56070144,\ (\ :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
13C1C_1 (1−T)2 ( 1 - T )^{2}
good5C22C_2^2 1+3T+8T2+3pT3+p2T4 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4}
7D4D_{4} 1−3T+12T2−3pT3+p2T4 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4}
11C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17D4D_{4} 1+3T+32T2+3pT3+p2T4 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4}
19C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
31D4D_{4} 1−6T+54T2−6pT3+p2T4 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−7T+48T2−7pT3+p2T4 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+6T+74T2+6pT3+p2T4 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+3T−18T2+3pT3+p2T4 1 + 3 T - 18 T^{2} + 3 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−9T+76T2−9pT3+p2T4 1 - 9 T + 76 T^{2} - 9 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+18T+170T2+18pT3+p2T4 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4}
59C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
61D4D_{4} 1+2T−30T2+2pT3+p2T4 1 + 2 T - 30 T^{2} + 2 p T^{3} + p^{2} T^{4}
67C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
71D4D_{4} 1+9T+124T2+9pT3+p2T4 1 + 9 T + 124 T^{2} + 9 p T^{3} + p^{2} T^{4}
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
83D4D_{4} 1+10T+38T2+10pT3+p2T4 1 + 10 T + 38 T^{2} + 10 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+110T2+p2T4 1 + 110 T^{2} + p^{2} T^{4}
97C2C_2 (1+6T+pT2)2 ( 1 + 6 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

−7.72094241232081977650438220022, −7.70780230892393800680692613034, −6.84183841404954672785359877157, −6.61281408280301419881473003117, −6.41271512730125236255735154944, −6.26151843122698450260910629957, −5.59396507919488738155589650106, −5.18100658843356690000311929060, −4.54646553995832739439507953528, −4.47087145005024048048879342982, −4.17737711598404645504832062210, −4.02941976587680069192120515041, −3.29507607954844955381963802382, −3.13938350290223621990433141581, −2.19700950997448984117901349121, −2.18857996435631769302187881544, −1.36065443931923485091698259288, −1.21867796196456251070705972321, 0, 0, 1.21867796196456251070705972321, 1.36065443931923485091698259288, 2.18857996435631769302187881544, 2.19700950997448984117901349121, 3.13938350290223621990433141581, 3.29507607954844955381963802382, 4.02941976587680069192120515041, 4.17737711598404645504832062210, 4.47087145005024048048879342982, 4.54646553995832739439507953528, 5.18100658843356690000311929060, 5.59396507919488738155589650106, 6.26151843122698450260910629957, 6.41271512730125236255735154944, 6.61281408280301419881473003117, 6.84183841404954672785359877157, 7.70780230892393800680692613034, 7.72094241232081977650438220022

Graph of the ZZ-function along the critical line