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Properties

Label 8-832e4-1.1-c1e4-0-19
Degree 88
Conductor 479174066176479174066176
Sign 11
Analytic cond. 1948.051948.05
Root an. cond. 2.577502.57750
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  + 10·9-s − 4·11-s − 8·13-s + 8·19-s + 16·23-s − 8·29-s + 8·31-s + 8·37-s + 8·41-s + 20·43-s − 8·47-s + 16·59-s − 8·61-s + 4·67-s − 24·73-s + 57·81-s − 36·83-s − 24·89-s − 28·97-s − 40·99-s + 8·103-s + 24·109-s + 64·113-s − 80·117-s + 8·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 10/3·9-s − 1.20·11-s − 2.21·13-s + 1.83·19-s + 3.33·23-s − 1.48·29-s + 1.43·31-s + 1.31·37-s + 1.24·41-s + 3.04·43-s − 1.16·47-s + 2.08·59-s − 1.02·61-s + 0.488·67-s − 2.80·73-s + 19/3·81-s − 3.95·83-s − 2.54·89-s − 2.84·97-s − 4.02·99-s + 0.788·103-s + 2.29·109-s + 6.02·113-s − 7.39·117-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

Λ(s)=((224⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅1342^{24} \cdot 13^{4}
Sign: 11
Analytic conductor: 1948.051948.05
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{24} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 4.7658977604.765897760
L(12)L(\frac12) ≈\approx 4.7658977604.765897760
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
13C22C_2^2 1+8T+32T2+8pT3+p2T4 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4}
good3C22C_2^2 (1−5T2+p2T4)2 ( 1 - 5 T^{2} + p^{2} T^{4} )^{2}
5C23C_2^3 1−p2T4+p4T8 1 - p^{2} T^{4} + p^{4} T^{8}
7C23C_2^3 1−17T4+p4T8 1 - 17 T^{4} + p^{4} T^{8}
11C2C_2×\timesC22C_2^2 (1+2T+pT2)2(1−18T2+p2T4) ( 1 + 2 T + p T^{2} )^{2}( 1 - 18 T^{2} + p^{2} T^{4} )
17D4×C2D_4\times C_2 1+14T2+467T4+14p2T6+p4T8 1 + 14 T^{2} + 467 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8}
19D4×C2D_4\times C_2 1−8T+32T2−56T3−46T4−56pT5+32p2T6−8p3T7+p4T8 1 - 8 T + 32 T^{2} - 56 T^{3} - 46 T^{4} - 56 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
23D4D_{4} (1−8T+52T2−8pT3+p2T4)2 ( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
29D4D_{4} (1+4T+52T2+4pT3+p2T4)2 ( 1 + 4 T + 52 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
31D4×C2D_4\times C_2 1−8T+32T2−152T3+578T4−152pT5+32p2T6−8p3T7+p4T8 1 - 8 T + 32 T^{2} - 152 T^{3} + 578 T^{4} - 152 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
37D4×C2D_4\times C_2 1−8T+32T2−320T3+3191T4−320pT5+32p2T6−8p3T7+p4T8 1 - 8 T + 32 T^{2} - 320 T^{3} + 3191 T^{4} - 320 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
41D4×C2D_4\times C_2 1−8T+32T2−232T3+1538T4−232pT5+32p2T6−8p3T7+p4T8 1 - 8 T + 32 T^{2} - 232 T^{3} + 1538 T^{4} - 232 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
43C2C_2 (1−5T+pT2)4 ( 1 - 5 T + p T^{2} )^{4}
47D4×C2D_4\times C_2 1+8T+32T2+80T3−1169T4+80pT5+32p2T6+8p3T7+p4T8 1 + 8 T + 32 T^{2} + 80 T^{3} - 1169 T^{4} + 80 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
53C22C_2^2 (1+16T2+p2T4)2 ( 1 + 16 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1−8T+32T2−8pT3+p2T4)2 ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
61C2C_2 (1+2T+pT2)4 ( 1 + 2 T + p T^{2} )^{4}
67D4×C2D_4\times C_2 1−4T+8T2+44T3−5842T4+44pT5+8p2T6−4p3T7+p4T8 1 - 4 T + 8 T^{2} + 44 T^{3} - 5842 T^{4} + 44 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
71C23C_2^3 1+8687T4+p4T8 1 + 8687 T^{4} + p^{4} T^{8}
73C22C_2^2 (1+12T+72T2+12pT3+p2T4)2 ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
79D4×C2D_4\times C_2 1−64T2+546T4−64p2T6+p4T8 1 - 64 T^{2} + 546 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8}
83D4×C2D_4\times C_2 1+36T+648T2+8100T3+81086T4+8100pT5+648p2T6+36p3T7+p4T8 1 + 36 T + 648 T^{2} + 8100 T^{3} + 81086 T^{4} + 8100 p T^{5} + 648 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8}
89D4×C2D_4\times C_2 1+24T+288T2+3384T3+37058T4+3384pT5+288p2T6+24p3T7+p4T8 1 + 24 T + 288 T^{2} + 3384 T^{3} + 37058 T^{4} + 3384 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}
97D4×C2D_4\times C_2 1+28T+392T2+4900T3+55166T4+4900pT5+392p2T6+28p3T7+p4T8 1 + 28 T + 392 T^{2} + 4900 T^{3} + 55166 T^{4} + 4900 p T^{5} + 392 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} 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

−7.24540711618315172626731583655, −7.18484291236185538987941836353, −7.02944431273268137596511653471, −6.93240619363106360047198192706, −6.76154769000676520323967908010, −5.89047693530968064178407972419, −5.78345553161091382466960814130, −5.76902237208018730887263107122, −5.68060530352411107430625892539, −5.01078488177627523989330790361, −4.79003191676737789047365590490, −4.70277011583786354655669338397, −4.56942151507851981626280540177, −4.41091918660200354684190613793, −4.06520034774877815047634847791, −3.76852924375064055021378055185, −3.11570543813214050581105733495, −3.10680322312971677808089268105, −2.86741804465341959484589053762, −2.46213595532823798708603104831, −2.22953969707436585094543426499, −1.76108011685486823386403432172, −1.17888513327873646033439619151, −1.10959431488628920150437446405, −0.63920390808653339447135156787, 0.63920390808653339447135156787, 1.10959431488628920150437446405, 1.17888513327873646033439619151, 1.76108011685486823386403432172, 2.22953969707436585094543426499, 2.46213595532823798708603104831, 2.86741804465341959484589053762, 3.10680322312971677808089268105, 3.11570543813214050581105733495, 3.76852924375064055021378055185, 4.06520034774877815047634847791, 4.41091918660200354684190613793, 4.56942151507851981626280540177, 4.70277011583786354655669338397, 4.79003191676737789047365590490, 5.01078488177627523989330790361, 5.68060530352411107430625892539, 5.76902237208018730887263107122, 5.78345553161091382466960814130, 5.89047693530968064178407972419, 6.76154769000676520323967908010, 6.93240619363106360047198192706, 7.02944431273268137596511653471, 7.18484291236185538987941836353, 7.24540711618315172626731583655

Graph of the ZZ-function along the critical line