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Properties

Label 8-700e4-1.1-c1e4-0-11
Degree 88
Conductor 240100000000240100000000
Sign 11
Analytic cond. 976.114976.114
Root an. cond. 2.364212.36421
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  + 3·9-s + 4·11-s − 12·29-s − 4·31-s + 20·41-s + 13·49-s − 16·59-s − 14·61-s + 32·71-s + 8·79-s + 9·81-s + 26·89-s + 12·99-s + 6·101-s + 18·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + ⋯
L(s)  = 1  + 9-s + 1.20·11-s − 2.22·29-s − 0.718·31-s + 3.12·41-s + 13/7·49-s − 2.08·59-s − 1.79·61-s + 3.79·71-s + 0.900·79-s + 81-s + 2.75·89-s + 1.20·99-s + 0.597·101-s + 1.72·109-s + 2.36·121-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 − 1.53·169-s + 0.0760·173-s + ⋯

Functional equation

Λ(s)=((28⋅58⋅74)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((28⋅58⋅74)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 28⋅58⋅742^{8} \cdot 5^{8} \cdot 7^{4}
Sign: 11
Analytic conductor: 976.114976.114
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 28⋅58⋅74, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 3.8554045303.855404530
L(12)L(\frac12) ≈\approx 3.8554045303.855404530
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
5 1 1
7C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
good3C2C_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 (1−2T−7T2−2pT3+p2T4)2 ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
13C2C_2 (1−4T+pT2)2(1+4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}
17C22C_2^2×\timesC22C_2^2 (1−8T+47T2−8pT3+p2T4)(1+8T+47T2+8pT3+p2T4) ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} )
19C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
23C23C_2^3 1−35T2+696T4−35p2T6+p4T8 1 - 35 T^{2} + 696 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8}
29C2C_2 (1+3T+pT2)4 ( 1 + 3 T + p T^{2} )^{4}
31C22C_2^2 (1+2T−27T2+2pT3+p2T4)2 ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
37C23C_2^3 1+10T2−1269T4+10p2T6+p4T8 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8}
41C2C_2 (1−5T+pT2)4 ( 1 - 5 T + p T^{2} )^{4}
43C22C_2^2 (1−85T2+p2T4)2 ( 1 - 85 T^{2} + p^{2} T^{4} )^{2}
47C23C_2^3 1+30T2−1309T4+30p2T6+p4T8 1 + 30 T^{2} - 1309 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8}
53C22C_2^2×\timesC22C_2^2 (1−14T+143T2−14pT3+p2T4)(1+14T+143T2+14pT3+p2T4) ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} )
59C22C_2^2 (1+8T+5T2+8pT3+p2T4)2 ( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
61C22C_2^2 (1+7T−12T2+7pT3+p2T4)2 ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2}
67C23C_2^3 1+125T2+11136T4+125p2T6+p4T8 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8}
71C2C_2 (1−8T+pT2)4 ( 1 - 8 T + p T^{2} )^{4}
73C23C_2^3 1−50T2−2829T4−50p2T6+p4T8 1 - 50 T^{2} - 2829 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8}
79C2C_2 (1−17T+pT2)2(1+13T+pT2)2 ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2}
83C22C_2^2 (1−165T2+p2T4)2 ( 1 - 165 T^{2} + p^{2} T^{4} )^{2}
89C22C_2^2 (1−13T+80T2−13pT3+p2T4)2 ( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2}
97C22C_2^2 (1−94T2+p2T4)2 ( 1 - 94 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

−7.63793255811454605600635851568, −7.31817762504316894269345615549, −7.17221391813796075269955755591, −6.71159370016484315171196704011, −6.70824588255331453991686156353, −6.18295792692781301525212610270, −6.17608890009838146258827061146, −5.88896691527138101467521251661, −5.84711342882042746286338130168, −5.18872742206798954139342115549, −5.18314219591605660247033744968, −4.87661561592478920467818965468, −4.59818711338232673591791562115, −4.15013238338767560023104257017, −4.06916457450381108752699357260, −3.78097153947895385409843541218, −3.72783207606035734724301947734, −3.11553387377460623112050030319, −3.10169379256012402887933854753, −2.37389021079181415499955452139, −2.21874116559189075748288815535, −1.72196293071766099245710907694, −1.70756452577681252597114183710, −0.825344116128527881505054229628, −0.71750846692866205821574959380, 0.71750846692866205821574959380, 0.825344116128527881505054229628, 1.70756452577681252597114183710, 1.72196293071766099245710907694, 2.21874116559189075748288815535, 2.37389021079181415499955452139, 3.10169379256012402887933854753, 3.11553387377460623112050030319, 3.72783207606035734724301947734, 3.78097153947895385409843541218, 4.06916457450381108752699357260, 4.15013238338767560023104257017, 4.59818711338232673591791562115, 4.87661561592478920467818965468, 5.18314219591605660247033744968, 5.18872742206798954139342115549, 5.84711342882042746286338130168, 5.88896691527138101467521251661, 6.17608890009838146258827061146, 6.18295792692781301525212610270, 6.70824588255331453991686156353, 6.71159370016484315171196704011, 7.17221391813796075269955755591, 7.31817762504316894269345615549, 7.63793255811454605600635851568

Graph of the ZZ-function along the critical line