↑

Properties

Label 4-384e2-1.1-c1e2-0-45
Degree 44
Conductor 147456147456
Sign −1-1
Analytic cond. 9.401929.40192
Root an. cond. 1.751071.75107
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 9-s − 4·17-s − 8·23-s − 2·25-s − 8·31-s − 4·41-s − 16·47-s − 10·49-s − 8·71-s + 12·73-s + 8·79-s + 81-s + 4·89-s + 4·97-s − 16·103-s + 4·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1/3·9-s − 0.970·17-s − 1.66·23-s − 2/5·25-s − 1.43·31-s − 0.624·41-s − 2.33·47-s − 1.42·49-s − 0.949·71-s + 1.40·73-s + 0.900·79-s + 1/9·81-s + 0.423·89-s + 0.406·97-s − 1.57·103-s + 0.376·113-s + 0.909·121-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 − 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

Λ(s)=(147456s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(147456s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 147456147456    =    214⋅322^{14} \cdot 3^{2}
Sign: −1-1
Analytic conductor: 9.401929.40192
Root analytic conductor: 1.751071.75107
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 147456, ( :1/2,1/2), −1)(4,\ 147456,\ (\ :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
3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good5C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
7C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
29C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1+2T+pT2)(1+6T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
37C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
41C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1+4T+pT2)(1+12T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
53C22C_2^2 1+82T2+p2T4 1 + 82 T^{2} + p^{2} T^{4}
59C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
61C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
67C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−14T+pT2)(1+2T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−10T+pT2)(1+2T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )
83C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
89C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
97C2C_2×\timesC2C_2 (1−18T+pT2)(1+14T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{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

−9.201358350412436276971805048079, −8.452978353281669605546395466725, −8.086442084037045849066002768206, −7.75550416430445217121413358077, −6.93561744728787393669725940046, −6.70362656258604102218812142163, −6.07594650208600870826189799266, −5.58468685813797179037662886736, −4.86181106786101780733721327564, −4.44564826696090424146474552868, −3.70892625990603304211839494755, −3.25898073626678173512567067932, −2.14369263635030970775219746009, −1.71140582351257440196866579310, 0, 1.71140582351257440196866579310, 2.14369263635030970775219746009, 3.25898073626678173512567067932, 3.70892625990603304211839494755, 4.44564826696090424146474552868, 4.86181106786101780733721327564, 5.58468685813797179037662886736, 6.07594650208600870826189799266, 6.70362656258604102218812142163, 6.93561744728787393669725940046, 7.75550416430445217121413358077, 8.086442084037045849066002768206, 8.452978353281669605546395466725, 9.201358350412436276971805048079

Graph of the ZZ-function along the critical line