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Properties

Label 8-2160e4-1.1-c2e4-0-4
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 1.19992×1071.19992\times 10^{7}
Root an. cond. 7.671747.67174
Motivic weight 22
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·5-s + 12·17-s + 12·19-s + 60·23-s + 66·25-s + 68·31-s − 240·47-s + 8·49-s − 204·53-s − 196·61-s − 180·79-s + 108·83-s − 144·85-s − 144·95-s − 144·107-s − 76·109-s − 48·113-s − 720·115-s + 96·121-s − 156·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 816·155-s + ⋯
L(s)  = 1  − 2.39·5-s + 0.705·17-s + 0.631·19-s + 2.60·23-s + 2.63·25-s + 2.19·31-s − 5.10·47-s + 8/49·49-s − 3.84·53-s − 3.21·61-s − 2.27·79-s + 1.30·83-s − 1.69·85-s − 1.51·95-s − 1.34·107-s − 0.697·109-s − 0.424·113-s − 6.26·115-s + 0.793·121-s − 1.24·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 5.26·155-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅312⋅542^{16} \cdot 3^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 1.19992×1071.19992\times 10^{7}
Root analytic conductor: 7.671747.67174
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :1,1,1,1), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.48061876080.4806187608
L(12)L(\frac12) ≈\approx 0.48061876080.4806187608
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
3 1 1
5D4D_{4} 1+12T+78T2+12p2T3+p4T4 1 + 12 T + 78 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4}
good7C22≀C2C_2^2 \wr C_2 1−8T2−3894T4−8p4T6+p8T8 1 - 8 T^{2} - 3894 T^{4} - 8 p^{4} T^{6} + p^{8} T^{8}
11C22≀C2C_2^2 \wr C_2 1−96T2+29786T4−96p4T6+p8T8 1 - 96 T^{2} + 29786 T^{4} - 96 p^{4} T^{6} + p^{8} T^{8}
13C22≀C2C_2^2 \wr C_2 1−352T2+71898T4−352p4T6+p8T8 1 - 352 T^{2} + 71898 T^{4} - 352 p^{4} T^{6} + p^{8} T^{8}
17D4D_{4} (1−6T+489T2−6p2T3+p4T4)2 ( 1 - 6 T + 489 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2}
19D4D_{4} (1−6T+713T2−6p2T3+p4T4)2 ( 1 - 6 T + 713 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2}
23D4D_{4} (1−30T+891T2−30p2T3+p4T4)2 ( 1 - 30 T + 891 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} )^{2}
29C22≀C2C_2^2 \wr C_2 1−1096T2+1242474T4−1096p4T6+p8T8 1 - 1096 T^{2} + 1242474 T^{4} - 1096 p^{4} T^{6} + p^{8} T^{8}
31D4D_{4} (1−34T+1761T2−34p2T3+p4T4)2 ( 1 - 34 T + 1761 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2}
37C22≀C2C_2^2 \wr C_2 1−68T2−1268634T4−68p4T6+p8T8 1 - 68 T^{2} - 1268634 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8}
41C22≀C2C_2^2 \wr C_2 1−5352T2+12407498T4−5352p4T6+p8T8 1 - 5352 T^{2} + 12407498 T^{4} - 5352 p^{4} T^{6} + p^{8} T^{8}
43C22≀C2C_2^2 \wr C_2 1−5672T2+14203050T4−5672p4T6+p8T8 1 - 5672 T^{2} + 14203050 T^{4} - 5672 p^{4} T^{6} + p^{8} T^{8}
47D4D_{4} (1+120T+7626T2+120p2T3+p4T4)2 ( 1 + 120 T + 7626 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} )^{2}
53D4D_{4} (1+102T+8057T2+102p2T3+p4T4)2 ( 1 + 102 T + 8057 T^{2} + 102 p^{2} T^{3} + p^{4} T^{4} )^{2}
59C22≀C2C_2^2 \wr C_2 1−10848T2+53616410T4−10848p4T6+p8T8 1 - 10848 T^{2} + 53616410 T^{4} - 10848 p^{4} T^{6} + p^{8} T^{8}
61D4D_{4} (1+98T+8691T2+98p2T3+p4T4)2 ( 1 + 98 T + 8691 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} )^{2}
67C22≀C2C_2^2 \wr C_2 1+6508T2+32310150T4+6508p4T6+p8T8 1 + 6508 T^{2} + 32310150 T^{4} + 6508 p^{4} T^{6} + p^{8} T^{8}
71C22≀C2C_2^2 \wr C_2 1−4288T2+45932730T4−4288p4T6+p8T8 1 - 4288 T^{2} + 45932730 T^{4} - 4288 p^{4} T^{6} + p^{8} T^{8}
73C22≀C2C_2^2 \wr C_2 1−13280T2+84777594T4−13280p4T6+p8T8 1 - 13280 T^{2} + 84777594 T^{4} - 13280 p^{4} T^{6} + p^{8} T^{8}
79D4D_{4} (1+90T+6569T2+90p2T3+p4T4)2 ( 1 + 90 T + 6569 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} )^{2}
83D4D_{4} (1−54T+7779T2−54p2T3+p4T4)2 ( 1 - 54 T + 7779 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} )^{2}
89C22≀C2C_2^2 \wr C_2 1−8136T2+91108874T4−8136p4T6+p8T8 1 - 8136 T^{2} + 91108874 T^{4} - 8136 p^{4} T^{6} + p^{8} T^{8}
97C22≀C2C_2^2 \wr C_2 1−19172T2+267913158T4−19172p4T6+p8T8 1 - 19172 T^{2} + 267913158 T^{4} - 19172 p^{4} T^{6} + p^{8} T^{8}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.26186041706497531726474535181, −6.12310179363891316749945228676, −5.88193141609136567394916415885, −5.69992144826141595842950022742, −5.14285688501537922992111360917, −5.09574616700765144850903279678, −5.00300765022317042222633314926, −4.55166912381640957759594387508, −4.52661808513471474313379342374, −4.45082910982069498577606907041, −4.42926075245793265168381229551, −3.62531406683303089254040972586, −3.45672483929030347647434243328, −3.43782493273158093036393690539, −3.30605098114644118486140066233, −3.02479279264904845119846223968, −2.87421408837980613356708234999, −2.58300709810755856131200400913, −2.17812806087464083816625959186, −1.63963185506767959032358117165, −1.26678775294253196977690566246, −1.21338092230257440535752697091, −1.14824841749661257816504019944, −0.25320905450733790111707806014, −0.20545440607039526451634565472, 0.20545440607039526451634565472, 0.25320905450733790111707806014, 1.14824841749661257816504019944, 1.21338092230257440535752697091, 1.26678775294253196977690566246, 1.63963185506767959032358117165, 2.17812806087464083816625959186, 2.58300709810755856131200400913, 2.87421408837980613356708234999, 3.02479279264904845119846223968, 3.30605098114644118486140066233, 3.43782493273158093036393690539, 3.45672483929030347647434243328, 3.62531406683303089254040972586, 4.42926075245793265168381229551, 4.45082910982069498577606907041, 4.52661808513471474313379342374, 4.55166912381640957759594387508, 5.00300765022317042222633314926, 5.09574616700765144850903279678, 5.14285688501537922992111360917, 5.69992144826141595842950022742, 5.88193141609136567394916415885, 6.12310179363891316749945228676, 6.26186041706497531726474535181

Graph of the ZZ-function along the critical line