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Properties

Label 8-1575e4-1.1-c1e4-0-1
Degree 88
Conductor 6.154×10126.154\times 10^{12}
Sign 11
Analytic cond. 25016.725016.7
Root an. cond. 3.546323.54632
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  + 2·4-s + 8·11-s + 3·16-s − 32·29-s − 8·41-s + 16·44-s − 2·49-s + 16·59-s + 24·61-s + 12·64-s + 8·71-s − 24·89-s + 40·101-s − 40·109-s − 64·116-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 16·164-s + 167-s + 28·169-s + ⋯
L(s)  = 1  + 4-s + 2.41·11-s + 3/4·16-s − 5.94·29-s − 1.24·41-s + 2.41·44-s − 2/7·49-s + 2.08·59-s + 3.07·61-s + 3/2·64-s + 0.949·71-s − 2.54·89-s + 3.98·101-s − 3.83·109-s − 5.94·116-s + 1.09·121-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 − 1.24·164-s + 0.0773·167-s + 2.15·169-s + ⋯

Functional equation

Λ(s)=((38⋅58⋅74)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{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)=((38⋅58⋅74)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{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: 38⋅58⋅743^{8} \cdot 5^{8} \cdot 7^{4}
Sign: 11
Analytic conductor: 25016.725016.7
Root analytic conductor: 3.546323.54632
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 38⋅58⋅74, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.74902752690.7490275269
L(12)L(\frac12) ≈\approx 0.74902752690.7490275269
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)
bad3 1 1
5 1 1
7C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
good2D4×C2D_4\times C_2 1−pT2+T4−p3T6+p4T8 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8}
11D4D_{4} (1−4T+18T2−4pT3+p2T4)2 ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
13D4×C2D_4\times C_2 1−28T2+406T4−28p2T6+p4T8 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8}
17C4×C2C_4\times C_2 1+4T2+70T4+4p2T6+p4T8 1 + 4 T^{2} + 70 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8}
19C22C_2^2 (1+30T2+p2T4)2 ( 1 + 30 T^{2} + p^{2} T^{4} )^{2}
23D4×C2D_4\times C_2 1−20T2+646T4−20p2T6+p4T8 1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}
29C2C_2 (1+8T+pT2)4 ( 1 + 8 T + p T^{2} )^{4}
31C22C_2^2 (1−10T2+p2T4)2 ( 1 - 10 T^{2} + p^{2} T^{4} )^{2}
37C22C_2^2 (1−38T2+p2T4)2 ( 1 - 38 T^{2} + p^{2} T^{4} )^{2}
41D4D_{4} (1+4T+54T2+4pT3+p2T4)2 ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1−76T2+3094T4−76p2T6+p4T8 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8}
47C22C_2^2 (1−78T2+p2T4)2 ( 1 - 78 T^{2} + p^{2} T^{4} )^{2}
53D4×C2D_4\times C_2 1−68T2+4726T4−68p2T6+p4T8 1 - 68 T^{2} + 4726 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8}
59C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
61C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
67D4×C2D_4\times C_2 1+20T2+886T4+20p2T6+p4T8 1 + 20 T^{2} + 886 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1−4T+74T2−4pT3+p2T4)2 ( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1+116T2+10822T4+116p2T6+p4T8 1 + 116 T^{2} + 10822 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8}
79C22C_2^2 (1+126T2+p2T4)2 ( 1 + 126 T^{2} + p^{2} T^{4} )^{2}
83C22C_2^2 (1−102T2+p2T4)2 ( 1 - 102 T^{2} + p^{2} T^{4} )^{2}
89C4C_4 (1+12T+86T2+12pT3+p2T4)2 ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−300T2+40166T4−300p2T6+p4T8 1 - 300 T^{2} + 40166 T^{4} - 300 p^{2} T^{6} + 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

−6.85329403574493223044627047580, −6.46427732159324793843380970929, −6.31360344724133285924808681745, −6.27973680996660126595340740270, −5.81607308120844197657008752173, −5.50007349180539755116699059936, −5.41517212914690456979566204726, −5.39180608208496519223926105476, −5.26962678418123954728303725645, −4.58633996598587717213865152763, −4.57439908541044768565528067486, −3.98968959882373509877399624182, −3.87221734926877695632809105934, −3.85375577704468597653801541684, −3.57633060670762066066954313065, −3.38448795743330742643513596726, −3.33188895437703631700515446817, −2.44846236965363946029852847867, −2.36683198387829876158923926619, −2.17290118948740712111499433321, −2.05219627483637834166423560279, −1.41633441648611835895558654391, −1.21098970600976099045330974551, −1.20701618278576920330468950728, −0.13883502307462636961419284135, 0.13883502307462636961419284135, 1.20701618278576920330468950728, 1.21098970600976099045330974551, 1.41633441648611835895558654391, 2.05219627483637834166423560279, 2.17290118948740712111499433321, 2.36683198387829876158923926619, 2.44846236965363946029852847867, 3.33188895437703631700515446817, 3.38448795743330742643513596726, 3.57633060670762066066954313065, 3.85375577704468597653801541684, 3.87221734926877695632809105934, 3.98968959882373509877399624182, 4.57439908541044768565528067486, 4.58633996598587717213865152763, 5.26962678418123954728303725645, 5.39180608208496519223926105476, 5.41517212914690456979566204726, 5.50007349180539755116699059936, 5.81607308120844197657008752173, 6.27973680996660126595340740270, 6.31360344724133285924808681745, 6.46427732159324793843380970929, 6.85329403574493223044627047580

Graph of the ZZ-function along the critical line