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Properties

Label 8-507e4-1.1-c1e4-0-5
Degree 88
Conductor 6607418840166074188401
Sign 11
Analytic cond. 268.621268.621
Root an. cond. 2.012062.01206
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  − 4·3-s − 4·7-s + 6·9-s − 16-s − 4·19-s + 16·21-s + 4·27-s + 20·31-s − 4·37-s + 4·48-s + 8·49-s + 16·57-s + 32·61-s − 24·63-s + 20·67-s − 4·73-s − 40·79-s − 37·81-s − 80·93-s − 28·97-s − 4·109-s + 16·111-s + 4·112-s + 127-s + 131-s + 16·133-s + 137-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.51·7-s + 2·9-s − 1/4·16-s − 0.917·19-s + 3.49·21-s + 0.769·27-s + 3.59·31-s − 0.657·37-s + 0.577·48-s + 8/7·49-s + 2.11·57-s + 4.09·61-s − 3.02·63-s + 2.44·67-s − 0.468·73-s − 4.50·79-s − 4.11·81-s − 8.29·93-s − 2.84·97-s − 0.383·109-s + 1.51·111-s + 0.377·112-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + 0.0854·137-s + ⋯

Functional equation

Λ(s)=((34⋅138)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((34⋅138)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅1383^{4} \cdot 13^{8}
Sign: 11
Analytic conductor: 268.621268.621
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅138, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.39600657400.3960065740
L(12)L(\frac12) ≈\approx 0.39600657400.3960065740
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)
bad3C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
13 1 1
good2C23C_2^3 1+T4+p4T8 1 + T^{4} + p^{4} T^{8}
5C22C_2^2×\timesC22C_2^2 (1−8T2+p2T4)(1+8T2+p2T4) ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} )
7C22C_2^2 (1+2T+2T2+2pT3+p2T4)2 ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
11C23C_2^3 1−206T4+p4T8 1 - 206 T^{4} + p^{4} T^{8}
17C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
19C22C_2^2 (1+2T+2T2+2pT3+p2T4)2 ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
23C22C_2^2 (1−26T2+p2T4)2 ( 1 - 26 T^{2} + p^{2} T^{4} )^{2}
29C22C_2^2 (1−50T2+p2T4)2 ( 1 - 50 T^{2} + p^{2} T^{4} )^{2}
31C22C_2^2 (1−10T+50T2−10pT3+p2T4)2 ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}
37C22C_2^2 (1+2T+2T2+2pT3+p2T4)2 ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
41C23C_2^3 1+2722T4+p4T8 1 + 2722 T^{4} + p^{4} T^{8}
43C22C_2^2 (1−50T2+p2T4)2 ( 1 - 50 T^{2} + p^{2} T^{4} )^{2}
47C23C_2^3 1+1666T4+p4T8 1 + 1666 T^{4} + p^{4} T^{8}
53C22C_2^2 (1−74T2+p2T4)2 ( 1 - 74 T^{2} + p^{2} T^{4} )^{2}
59C23C_2^3 1+3442T4+p4T8 1 + 3442 T^{4} + p^{4} T^{8}
61C2C_2 (1−8T+pT2)4 ( 1 - 8 T + p T^{2} )^{4}
67C22C_2^2 (1−10T+50T2−10pT3+p2T4)2 ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}
71C23C_2^3 1+5794T4+p4T8 1 + 5794 T^{4} + p^{4} T^{8}
73C22C_2^2 (1+2T+2T2+2pT3+p2T4)2 ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
79C2C_2 (1+10T+pT2)4 ( 1 + 10 T + p T^{2} )^{4}
83C23C_2^3 1−3374T4+p4T8 1 - 3374 T^{4} + p^{4} T^{8}
89C23C_2^3 1−15518T4+p4T8 1 - 15518 T^{4} + p^{4} T^{8}
97C22C_2^2 (1+14T+98T2+14pT3+p2T4)2 ( 1 + 14 T + 98 T^{2} + 14 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

−7.83822947520880574577364654393, −7.59181066812645919498436541365, −7.14141003972742684763235584438, −6.75899751472245250071524594636, −6.75660616510535113707480455451, −6.74557801224362246443472253505, −6.53910000091238888954378893823, −6.16030917518269674007320590691, −5.85971717573553252982609541817, −5.69794730910731047127424763801, −5.53266345040913408084528041862, −5.25512262109101644874096447763, −4.94803908289710508373848657373, −4.60827072143380032191079711651, −4.42871004452897072754881790645, −4.09101466394133302183611782626, −3.86625601007857624888927256146, −3.49778355333802456078703988908, −2.86027029231990694911627953365, −2.73324232175376469880236674273, −2.66679232346809366379394973186, −1.99684177936964598146125445298, −1.28217490816909952836313307976, −0.825172695563514558197423061919, −0.35295455859912775921321378120, 0.35295455859912775921321378120, 0.825172695563514558197423061919, 1.28217490816909952836313307976, 1.99684177936964598146125445298, 2.66679232346809366379394973186, 2.73324232175376469880236674273, 2.86027029231990694911627953365, 3.49778355333802456078703988908, 3.86625601007857624888927256146, 4.09101466394133302183611782626, 4.42871004452897072754881790645, 4.60827072143380032191079711651, 4.94803908289710508373848657373, 5.25512262109101644874096447763, 5.53266345040913408084528041862, 5.69794730910731047127424763801, 5.85971717573553252982609541817, 6.16030917518269674007320590691, 6.53910000091238888954378893823, 6.74557801224362246443472253505, 6.75660616510535113707480455451, 6.75899751472245250071524594636, 7.14141003972742684763235584438, 7.59181066812645919498436541365, 7.83822947520880574577364654393

Graph of the ZZ-function along the critical line