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Properties

Label 8-91e4-1.1-c1e4-0-1
Degree 88
Conductor 6857496168574961
Sign 11
Analytic cond. 0.2787870.278787
Root an. cond. 0.8524310.852431
Motivic weight 11
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  + 3·2-s + 5·4-s − 8·7-s + 6·8-s + 9-s + 6·11-s − 4·13-s − 24·14-s + 4·16-s + 6·17-s + 3·18-s − 6·19-s + 18·22-s + 12·23-s + 5·25-s − 12·26-s − 40·28-s − 10·31-s + 18·34-s + 5·36-s + 4·37-s − 18·38-s − 32·43-s + 30·44-s + 36·46-s + 6·47-s + 34·49-s + ⋯
L(s)  = 1  + 2.12·2-s + 5/2·4-s − 3.02·7-s + 2.12·8-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 6.41·14-s + 16-s + 1.45·17-s + 0.707·18-s − 1.37·19-s + 3.83·22-s + 2.50·23-s + 25-s − 2.35·26-s − 7.55·28-s − 1.79·31-s + 3.08·34-s + 5/6·36-s + 0.657·37-s − 2.91·38-s − 4.87·43-s + 4.52·44-s + 5.30·46-s + 0.875·47-s + 34/7·49-s + ⋯

Functional equation

Λ(s)=(68574961s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(68574961s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 88
Conductor: 6857496168574961    =    74⋅1347^{4} \cdot 13^{4}
Sign: 11
Analytic conductor: 0.2787870.278787
Root analytic conductor: 0.8524310.852431
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 68574961, ( :1/2,1/2,1/2,1/2), 1)(8,\ 68574961,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1269271752.126927175
L(12)L(\frac12) ≈\approx 2.1269271752.126927175
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)
bad7C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
13C1C_1 (1+T)4 ( 1 + T )^{4}
good2C22C_2^2×\timesC22C_2^2 (1−3T+5T2−3pT3+p2T4)(1−T2+p2T4) ( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 - T^{2} + p^{2} T^{4} )
3C23C_2^3 1−T2−8T4−p2T6+p4T8 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8}
5C2C_2×\timesC22C_2^2 (1−pT2)2(1+pT2+p2T4) ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} )
11C22C_2^2 (1−3T−2T2−3pT3+p2T4)2 ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
17D4×C2D_4\times C_2 1−6T+13T2+66T3−372T4+66pT5+13p2T6−6p3T7+p4T8 1 - 6 T + 13 T^{2} + 66 T^{3} - 372 T^{4} + 66 p T^{5} + 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
19C22C_2^2 (1+3T−10T2+3pT3+p2T4)2 ( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}
23D4×C2D_4\times C_2 1−12T+67T2−372T3+2088T4−372pT5+67p2T6−12p3T7+p4T8 1 - 12 T + 67 T^{2} - 372 T^{3} + 2088 T^{4} - 372 p T^{5} + 67 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}
29C22C_2^2 (1+38T2+p2T4)2 ( 1 + 38 T^{2} + p^{2} T^{4} )^{2}
31C22C_2^2 (1+5T−6T2+5pT3+p2T4)2 ( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1−4T−17T2+164T3−872T4+164pT5−17p2T6−4p3T7+p4T8 1 - 4 T - 17 T^{2} + 164 T^{3} - 872 T^{4} + 164 p T^{5} - 17 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
41C22C_2^2 (1+62T2+p2T4)2 ( 1 + 62 T^{2} + p^{2} T^{4} )^{2}
43C2C_2 (1+8T+pT2)4 ( 1 + 8 T + p T^{2} )^{4}
47D4×C2D_4\times C_2 1−6T−pT2+66T3+2988T4+66pT5−p3T6−6p3T7+p4T8 1 - 6 T - p T^{2} + 66 T^{3} + 2988 T^{4} + 66 p T^{5} - p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
53D4×C2D_4\times C_2 1−6T−59T2+66T3+4308T4+66pT5−59p2T6−6p3T7+p4T8 1 - 6 T - 59 T^{2} + 66 T^{3} + 4308 T^{4} + 66 p T^{5} - 59 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
59D4×C2D_4\times C_2 1+6T−71T2−66T3+5844T4−66pT5−71p2T6+6p3T7+p4T8 1 + 6 T - 71 T^{2} - 66 T^{3} + 5844 T^{4} - 66 p T^{5} - 71 p^{2} T^{6} + 6 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}
67C22C_2^2 (1−3T−58T2−3pT3+p2T4)2 ( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
71C22C_2^2 (1+62T2+p2T4)2 ( 1 + 62 T^{2} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1+8T−53T2−232T3+3688T4−232pT5−53p2T6+8p3T7+p4T8 1 + 8 T - 53 T^{2} - 232 T^{3} + 3688 T^{4} - 232 p T^{5} - 53 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
79D4×C2D_4\times C_2 1+8T−65T2−232T3+5344T4−232pT5−65p2T6+8p3T7+p4T8 1 + 8 T - 65 T^{2} - 232 T^{3} + 5344 T^{4} - 232 p T^{5} - 65 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
83C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
89C23C_2^3 1−173T2+22008T4−173p2T6+p4T8 1 - 173 T^{2} + 22008 T^{4} - 173 p^{2} T^{6} + p^{4} T^{8}
97D4D_{4} (1+8T+30T2+8pT3+p2T4)2 ( 1 + 8 T + 30 T^{2} + 8 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

−10.41507271308888056682771179061, −10.19511879307132675294626325601, −9.963985434852701662601982436292, −9.576703550189450260791544118253, −9.365255247777837370764012818202, −9.025745998523202722389118705428, −8.946273652550477929618178825074, −8.458333016419311493913696830101, −7.974674709273874811639719904103, −7.28707580737416211507804133607, −7.00073560932273874586189697192, −6.98263698841547473536537244743, −6.77337655772861857255230814779, −6.37522657177174766914774324039, −5.96738277904419845478407978227, −5.94049446538056836979371913842, −5.09779491066717041247569614310, −5.05436121462349775984850677814, −4.64406321105409139521068003200, −3.97646047461772695807843743915, −3.77734419038093641113834944118, −3.35327604878179764201177913968, −3.00255312100191105218792907301, −2.82081953332938905818375982573, −1.69057622283079173193073362078, 1.69057622283079173193073362078, 2.82081953332938905818375982573, 3.00255312100191105218792907301, 3.35327604878179764201177913968, 3.77734419038093641113834944118, 3.97646047461772695807843743915, 4.64406321105409139521068003200, 5.05436121462349775984850677814, 5.09779491066717041247569614310, 5.94049446538056836979371913842, 5.96738277904419845478407978227, 6.37522657177174766914774324039, 6.77337655772861857255230814779, 6.98263698841547473536537244743, 7.00073560932273874586189697192, 7.28707580737416211507804133607, 7.974674709273874811639719904103, 8.458333016419311493913696830101, 8.946273652550477929618178825074, 9.025745998523202722389118705428, 9.365255247777837370764012818202, 9.576703550189450260791544118253, 9.963985434852701662601982436292, 10.19511879307132675294626325601, 10.41507271308888056682771179061

Graph of the ZZ-function along the critical line