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Properties

Label 8-2156e4-1.1-c1e4-0-6
Degree 88
Conductor 2.161×10132.161\times 10^{13}
Sign 11
Analytic cond. 87842.287842.2
Root an. cond. 4.149184.14918
Motivic weight 11
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  + 4·9-s + 2·11-s + 8·23-s + 10·25-s + 8·29-s + 8·37-s + 8·43-s + 8·53-s + 16·67-s + 20·79-s + 9·81-s + 8·99-s + 24·107-s + 20·109-s + 56·113-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s + ⋯
L(s)  = 1  + 4/3·9-s + 0.603·11-s + 1.66·23-s + 2·25-s + 1.48·29-s + 1.31·37-s + 1.21·43-s + 1.09·53-s + 1.95·67-s + 2.25·79-s + 81-s + 0.804·99-s + 2.32·107-s + 1.91·109-s + 5.26·113-s + 1/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.69·169-s + 0.0760·173-s + ⋯

Functional equation

Λ(s)=((28⋅78⋅114)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((28⋅78⋅114)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 28⋅78⋅1142^{8} \cdot 7^{8} \cdot 11^{4}
Sign: 11
Analytic conductor: 87842.287842.2
Root analytic conductor: 4.149184.14918
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 28⋅78⋅114, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{8} \cdot 7^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 10.3113481410.31134814
L(12)L(\frac12) ≈\approx 10.3113481410.31134814
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
7 1 1
11C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
good3C23C_2^3 1−4T2+7T4−4p2T6+p4T8 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}
5C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
13C22C_2^2 (1+24T2+p2T4)2 ( 1 + 24 T^{2} + p^{2} T^{4} )^{2}
17C23C_2^3 1+16T2−33T4+16p2T6+p4T8 1 + 16 T^{2} - 33 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8}
19C23C_2^3 1−30T2+539T4−30p2T6+p4T8 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8}
23C22C_2^2 (1−4T−7T2−4pT3+p2T4)2 ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
29C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
31C23C_2^3 1−44T2+975T4−44p2T6+p4T8 1 - 44 T^{2} + 975 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8}
37C22C_2^2 (1−4T−21T2−4pT3+p2T4)2 ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
41C22C_2^2 (1+80T2+p2T4)2 ( 1 + 80 T^{2} + p^{2} T^{4} )^{2}
43C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
47C23C_2^3 1+4T2−2193T4+4p2T6+p4T8 1 + 4 T^{2} - 2193 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8}
53C22C_2^2 (1−4T−37T2−4pT3+p2T4)2 ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
59C23C_2^3 1−100T2+6519T4−100p2T6+p4T8 1 - 100 T^{2} + 6519 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8}
61C23C_2^3 1+40T2−2121T4+40p2T6+p4T8 1 + 40 T^{2} - 2121 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8}
67C22C_2^2 (1−8T−3T2−8pT3+p2T4)2 ( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
71C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
73C23C_2^3 1−144T2+15407T4−144p2T6+p4T8 1 - 144 T^{2} + 15407 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8}
79C22C_2^2 (1−10T+21T2−10pT3+p2T4)2 ( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}
83C22C_2^2 (1+94T2+p2T4)2 ( 1 + 94 T^{2} + p^{2} T^{4} )^{2}
89C23C_2^3 1−50T2−5421T4−50p2T6+p4T8 1 - 50 T^{2} - 5421 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8}
97C22C_2^2 (1+122T2+p2T4)2 ( 1 + 122 T^{2} + p^{2} T^{4} )^{2}
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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

−6.61029443398192901305017912042, −6.19913271860185442274621953720, −6.12499582334188336426202107904, −5.83802496510041860472718140662, −5.73895242078462659290465345690, −5.18517289328380608426318316847, −5.10263312365293997878856193351, −5.00481534702537304736312482120, −4.82547367399436220357001523186, −4.43854773878332127387637844526, −4.24742316397698469615886274129, −4.16312006777725377835686032411, −4.13894262629125312046146407048, −3.31553234841752865066648180006, −3.24999023558210615085528883730, −3.24532167763463205435077415999, −3.18501656123145836384266699957, −2.41499464965090300152443686101, −2.34704173477180368403423305598, −2.07962606377655548064038089994, −1.91967395231479030561324067062, −1.20921582178916836626915820982, −0.984167329389546213044403349925, −0.810889429789893705305067641421, −0.72919177675003228043815857496, 0.72919177675003228043815857496, 0.810889429789893705305067641421, 0.984167329389546213044403349925, 1.20921582178916836626915820982, 1.91967395231479030561324067062, 2.07962606377655548064038089994, 2.34704173477180368403423305598, 2.41499464965090300152443686101, 3.18501656123145836384266699957, 3.24532167763463205435077415999, 3.24999023558210615085528883730, 3.31553234841752865066648180006, 4.13894262629125312046146407048, 4.16312006777725377835686032411, 4.24742316397698469615886274129, 4.43854773878332127387637844526, 4.82547367399436220357001523186, 5.00481534702537304736312482120, 5.10263312365293997878856193351, 5.18517289328380608426318316847, 5.73895242078462659290465345690, 5.83802496510041860472718140662, 6.12499582334188336426202107904, 6.19913271860185442274621953720, 6.61029443398192901305017912042

Graph of the ZZ-function along the critical line