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Properties

Label 4-416e2-1.1-c2e2-0-2
Degree 44
Conductor 173056173056
Sign 11
Analytic cond. 128.486128.486
Root an. cond. 3.366773.36677
Motivic weight 22
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 18·9-s − 24·13-s + 2·25-s − 80·29-s − 94·37-s + 62·41-s − 36·45-s − 180·53-s + 44·61-s − 48·65-s − 14·73-s + 243·81-s + 238·89-s − 14·97-s − 302·109-s + 448·113-s + 432·117-s + 50·125-s + 127-s + 131-s + 137-s + 139-s − 160·145-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 2/5·5-s − 2·9-s − 1.84·13-s + 2/25·25-s − 2.75·29-s − 2.54·37-s + 1.51·41-s − 4/5·45-s − 3.39·53-s + 0.721·61-s − 0.738·65-s − 0.191·73-s + 3·81-s + 2.67·89-s − 0.144·97-s − 2.77·109-s + 3.96·113-s + 3.69·117-s + 2/5·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.10·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + ⋯

Functional equation

Λ(s)=(173056s/2 ΓC(s)2 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(173056s/2 ΓC(s+1)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 173056173056    =    210⋅1322^{10} \cdot 13^{2}
Sign: 11
Analytic conductor: 128.486128.486
Root analytic conductor: 3.366773.36677
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 173056, ( :1,1), 1)(4,\ 173056,\ (\ :1, 1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.44672956290.4467295629
L(12)L(\frac12) ≈\approx 0.44672956290.4467295629
L(2)L(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
13C2C_2 1+24T+p2T2 1 + 24 T + p^{2} T^{2}
good3C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
5C2C_2 (1−8T+p2T2)(1+6T+p2T2) ( 1 - 8 T + p^{2} T^{2} )( 1 + 6 T + p^{2} T^{2} )
7C22C_2^2 1+p4T4 1 + p^{4} T^{4}
11C22C_2^2 1+p4T4 1 + p^{4} T^{4}
17C2C_2 (1−16T+p2T2)(1+16T+p2T2) ( 1 - 16 T + p^{2} T^{2} )( 1 + 16 T + p^{2} T^{2} )
19C22C_2^2 1+p4T4 1 + p^{4} T^{4}
23C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
29C2C_2 (1+40T+p2T2)2 ( 1 + 40 T + p^{2} T^{2} )^{2}
31C22C_2^2 1+p4T4 1 + p^{4} T^{4}
37C2C_2 (1+24T+p2T2)(1+70T+p2T2) ( 1 + 24 T + p^{2} T^{2} )( 1 + 70 T + p^{2} T^{2} )
41C2C_2 (1−80T+p2T2)(1+18T+p2T2) ( 1 - 80 T + p^{2} T^{2} )( 1 + 18 T + p^{2} T^{2} )
43C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
47C22C_2^2 1+p4T4 1 + p^{4} T^{4}
53C2C_2 (1+90T+p2T2)2 ( 1 + 90 T + p^{2} T^{2} )^{2}
59C22C_2^2 1+p4T4 1 + p^{4} T^{4}
61C2C_2 (1−22T+p2T2)2 ( 1 - 22 T + p^{2} T^{2} )^{2}
67C22C_2^2 1+p4T4 1 + p^{4} T^{4}
71C22C_2^2 1+p4T4 1 + p^{4} T^{4}
73C2C_2 (1−96T+p2T2)(1+110T+p2T2) ( 1 - 96 T + p^{2} T^{2} )( 1 + 110 T + p^{2} T^{2} )
79C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
83C22C_2^2 1+p4T4 1 + p^{4} T^{4}
89C2C_2 (1−160T+p2T2)(1−78T+p2T2) ( 1 - 160 T + p^{2} T^{2} )( 1 - 78 T + p^{2} T^{2} )
97C2C_2 (1−130T+p2T2)(1+144T+p2T2) ( 1 - 130 T + p^{2} T^{2} )( 1 + 144 T + p^{2} T^{2} )
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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

−11.29905166719647095589353030257, −10.75173039444051713424636902245, −10.48614050325251223733904924575, −9.569656809423703145810550706156, −9.509207406736478001463496967803, −9.054678408906239659996246015826, −8.582645852115927996849715601487, −7.78175845676634739206104981931, −7.76871600921853883001956753435, −7.02431749490582890796766831814, −6.52194547920162278737915447252, −5.74970147081745967007595589089, −5.60200172361185307304694701074, −5.04099318443943936845259117297, −4.52508187023548470730925339140, −3.42722189233929092981039472361, −3.21017195142853361201540508793, −2.27040762533274432061971840140, −1.92523458226620130090730682705, −0.26482182830635757392643846210, 0.26482182830635757392643846210, 1.92523458226620130090730682705, 2.27040762533274432061971840140, 3.21017195142853361201540508793, 3.42722189233929092981039472361, 4.52508187023548470730925339140, 5.04099318443943936845259117297, 5.60200172361185307304694701074, 5.74970147081745967007595589089, 6.52194547920162278737915447252, 7.02431749490582890796766831814, 7.76871600921853883001956753435, 7.78175845676634739206104981931, 8.582645852115927996849715601487, 9.054678408906239659996246015826, 9.509207406736478001463496967803, 9.569656809423703145810550706156, 10.48614050325251223733904924575, 10.75173039444051713424636902245, 11.29905166719647095589353030257

Graph of the ZZ-function along the critical line