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Properties

Label 8-2160e4-1.1-c1e4-0-19
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 88495.988495.9
Root an. cond. 4.153034.15303
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  − 2·5-s + 2·7-s + 8·17-s + 8·19-s + 10·23-s + 25-s + 6·29-s + 12·31-s − 4·35-s − 24·37-s − 10·41-s + 4·43-s + 14·47-s + 9·49-s + 8·53-s + 12·59-s + 6·61-s − 2·67-s + 16·73-s − 4·79-s + 6·83-s − 16·85-s + 28·89-s − 16·95-s − 4·97-s − 4·101-s + 20·103-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s + 1.94·17-s + 1.83·19-s + 2.08·23-s + 1/5·25-s + 1.11·29-s + 2.15·31-s − 0.676·35-s − 3.94·37-s − 1.56·41-s + 0.609·43-s + 2.04·47-s + 9/7·49-s + 1.09·53-s + 1.56·59-s + 0.768·61-s − 0.244·67-s + 1.87·73-s − 0.450·79-s + 0.658·83-s − 1.73·85-s + 2.96·89-s − 1.64·95-s − 0.406·97-s − 0.398·101-s + 1.97·103-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−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(2-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s+1/2)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+1/2)^{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: 88495.988495.9
Root analytic conductor: 4.153034.15303
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 7.4515100797.451510079
L(12)L(\frac12) ≈\approx 7.4515100797.451510079
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
5C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
good7D4×C2D_4\times C_2 1−2T−5T2+10T3+4T4+10pT5−5p2T6−2p3T7+p4T8 1 - 2 T - 5 T^{2} + 10 T^{3} + 4 T^{4} + 10 p T^{5} - 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
11C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
13C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
17C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
19D4D_{4} (1−4T+18T2−4pT3+p2T4)2 ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
23D4×C2D_4\times C_2 1−10T+35T2−190T3+1396T4−190pT5+35p2T6−10p3T7+p4T8 1 - 10 T + 35 T^{2} - 190 T^{3} + 1396 T^{4} - 190 p T^{5} + 35 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1−6T−7T2+90T3−36T4+90pT5−7p2T6−6p3T7+p4T8 1 - 6 T - 7 T^{2} + 90 T^{3} - 36 T^{4} + 90 p T^{5} - 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
31D4×C2D_4\times C_2 1−12T+70T2−144T3+51T4−144pT5+70p2T6−12p3T7+p4T8 1 - 12 T + 70 T^{2} - 144 T^{3} + 51 T^{4} - 144 p T^{5} + 70 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}
37C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
41D4×C2D_4\times C_2 1+10T+17T2+10T3+1108T4+10pT5+17p2T6+10p3T7+p4T8 1 + 10 T + 17 T^{2} + 10 T^{3} + 1108 T^{4} + 10 p T^{5} + 17 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}
43D4×C2D_4\times C_2 1−4T+22T2+368T3−2501T4+368pT5+22p2T6−4p3T7+p4T8 1 - 4 T + 22 T^{2} + 368 T^{3} - 2501 T^{4} + 368 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
47D4×C2D_4\times C_2 1−14T+59T2−602T3+7348T4−602pT5+59p2T6−14p3T7+p4T8 1 - 14 T + 59 T^{2} - 602 T^{3} + 7348 T^{4} - 602 p T^{5} + 59 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}
53D4D_{4} (1−4T+14T2−4pT3+p2T4)2 ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1−12T+14T2−144T3+4923T4−144pT5+14p2T6−12p3T7+p4T8 1 - 12 T + 14 T^{2} - 144 T^{3} + 4923 T^{4} - 144 p T^{5} + 14 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}
61C22C_2^2 (1−3T−52T2−3pT3+p2T4)2 ( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1+2T+19T2−298T3−4532T4−298pT5+19p2T6+2p3T7+p4T8 1 + 2 T + 19 T^{2} - 298 T^{3} - 4532 T^{4} - 298 p T^{5} + 19 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
71C22C_2^2 (1+46T2+p2T4)2 ( 1 + 46 T^{2} + p^{2} T^{4} )^{2}
73D4D_{4} (1−8T+66T2−8pT3+p2T4)2 ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
79D4×C2D_4\times C_2 1+4T−122T2−80T3+11539T4−80pT5−122p2T6+4p3T7+p4T8 1 + 4 T - 122 T^{2} - 80 T^{3} + 11539 T^{4} - 80 p T^{5} - 122 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
83D4×C2D_4\times C_2 1−6T−133T2−18T3+18684T4−18pT5−133p2T6−6p3T7+p4T8 1 - 6 T - 133 T^{2} - 18 T^{3} + 18684 T^{4} - 18 p T^{5} - 133 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
89D4D_{4} (1−14T+131T2−14pT3+p2T4)2 ( 1 - 14 T + 131 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}
97C22C_2^2 (1+2T−93T2+2pT3+p2T4)2 ( 1 + 2 T - 93 T^{2} + 2 p T^{3} + 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

−6.75995113304210617087790432692, −6.01018507917860200364504927600, −6.00871199429714224143126170022, −5.91736405618493464233945516111, −5.38295945371299781371473888417, −5.24118409537129901167083672276, −5.18278829129028159646251404451, −5.15058730198881321357027395488, −4.91083687527932679981700741628, −4.50241615920619079438972761730, −4.35414153523709184134041458416, −4.00146965476361049211770660167, −3.89550821779315251172974022675, −3.35803792884046277530537631238, −3.35257568424756317001976607024, −3.20358041404664131716185968232, −3.20147267838816799584546743316, −2.54460583341413070962077348698, −2.38361779170187557576398274131, −2.12651015412862400858851335802, −1.72929136274523278830976506216, −1.22612582279908290900198994644, −1.04409572270346864466305798018, −0.77632012352639139228655465572, −0.61220779306442116758033154828, 0.61220779306442116758033154828, 0.77632012352639139228655465572, 1.04409572270346864466305798018, 1.22612582279908290900198994644, 1.72929136274523278830976506216, 2.12651015412862400858851335802, 2.38361779170187557576398274131, 2.54460583341413070962077348698, 3.20147267838816799584546743316, 3.20358041404664131716185968232, 3.35257568424756317001976607024, 3.35803792884046277530537631238, 3.89550821779315251172974022675, 4.00146965476361049211770660167, 4.35414153523709184134041458416, 4.50241615920619079438972761730, 4.91083687527932679981700741628, 5.15058730198881321357027395488, 5.18278829129028159646251404451, 5.24118409537129901167083672276, 5.38295945371299781371473888417, 5.91736405618493464233945516111, 6.00871199429714224143126170022, 6.01018507917860200364504927600, 6.75995113304210617087790432692

Graph of the ZZ-function along the critical line