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Properties

Label 8-1575e4-1.1-c1e4-0-14
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  + 4-s + 4·11-s + 4·16-s + 4·19-s − 8·29-s + 32·41-s + 4·44-s − 2·49-s − 12·59-s − 8·61-s + 11·64-s + 20·71-s + 4·76-s − 16·79-s + 4·89-s + 44·101-s − 28·109-s − 8·116-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.20·11-s + 16-s + 0.917·19-s − 1.48·29-s + 4.99·41-s + 0.603·44-s − 2/7·49-s − 1.56·59-s − 1.02·61-s + 11/8·64-s + 2.37·71-s + 0.458·76-s − 1.80·79-s + 0.423·89-s + 4.37·101-s − 2.68·109-s − 0.742·116-s + 6/11·121-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 + ⋯

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 6.7177839376.717783937
L(12)L(\frac12) ≈\approx 6.7177839376.717783937
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}
good2C23C_2^3 1−T2−3T4−p2T6+p4T8 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8}
11D4D_{4} (1−2T+3T2−2pT3+p2T4)2 ( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
13D4×C2D_4\times C_2 1−24T2+302T4−24p2T6+p4T8 1 - 24 T^{2} + 302 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8}
17D4×C2D_4\times C_2 1+24T2+542T4+24p2T6+p4T8 1 + 24 T^{2} + 542 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8}
19D4D_{4} (1−2T+34T2−2pT3+p2T4)2 ( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
23C22C_2^2 (1−21T2+p2T4)2 ( 1 - 21 T^{2} + p^{2} T^{4} )^{2}
29D4D_{4} (1+4T+17T2+4pT3+p2T4)2 ( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
31C22C_2^2 (1+42T2+p2T4)2 ( 1 + 42 T^{2} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1−106T2+5467T4−106p2T6+p4T8 1 - 106 T^{2} + 5467 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8}
41C2C_2 (1−8T+pT2)4 ( 1 - 8 T + p T^{2} )^{4}
43D4×C2D_4\times C_2 1−90T2+5003T4−90p2T6+p4T8 1 - 90 T^{2} + 5003 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8}
47D4×C2D_4\times C_2 1−128T2+8014T4−128p2T6+p4T8 1 - 128 T^{2} + 8014 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8}
53D4×C2D_4\times C_2 1−140T2+9238T4−140p2T6+p4T8 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1+6T+122T2+6pT3+p2T4)2 ( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
61C4C_4 (1+4T−54T2+4pT3+p2T4)2 ( 1 + 4 T - 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−146T2+11427T4−146p2T6+p4T8 1 - 146 T^{2} + 11427 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1−10T+147T2−10pT3+p2T4)2 ( 1 - 10 T + 147 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1−280T2+30238T4−280p2T6+p4T8 1 - 280 T^{2} + 30238 T^{4} - 280 p^{2} T^{6} + p^{4} T^{8}
79D4D_{4} (1+8T+169T2+8pT3+p2T4)2 ( 1 + 8 T + 169 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1−164T2+15382T4−164p2T6+p4T8 1 - 164 T^{2} + 15382 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8}
89D4D_{4} (1−2T+134T2−2pT3+p2T4)2 ( 1 - 2 T + 134 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−220T2+25798T4−220p2T6+p4T8 1 - 220 T^{2} + 25798 T^{4} - 220 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

−6.66775877760117130316455019265, −6.45480615872481844068539469859, −6.34974036175935273526693388067, −5.92866862813531010306566049673, −5.87415673104109862394886780134, −5.66944446697626137799834121124, −5.63997373239414771027463612164, −5.34859652009435925288220791751, −4.78898438152016987874951457191, −4.78525810740197747572049804826, −4.46162774579540810516331311285, −4.34571021059678637956435594192, −3.84825982488302644449214904540, −3.80661502230138959441136690908, −3.72067825028549213059396977759, −3.14145689669036212522323250713, −2.98185496620450859873366681031, −2.95901320591853251336578272933, −2.46882613788769181569860949530, −2.09217666317653287541762538032, −1.77616513749157110934777767466, −1.76052177084507313596685554946, −1.05143298270179940859915631036, −0.878081353727540340863022067729, −0.56978616255180607244449838896, 0.56978616255180607244449838896, 0.878081353727540340863022067729, 1.05143298270179940859915631036, 1.76052177084507313596685554946, 1.77616513749157110934777767466, 2.09217666317653287541762538032, 2.46882613788769181569860949530, 2.95901320591853251336578272933, 2.98185496620450859873366681031, 3.14145689669036212522323250713, 3.72067825028549213059396977759, 3.80661502230138959441136690908, 3.84825982488302644449214904540, 4.34571021059678637956435594192, 4.46162774579540810516331311285, 4.78525810740197747572049804826, 4.78898438152016987874951457191, 5.34859652009435925288220791751, 5.63997373239414771027463612164, 5.66944446697626137799834121124, 5.87415673104109862394886780134, 5.92866862813531010306566049673, 6.34974036175935273526693388067, 6.45480615872481844068539469859, 6.66775877760117130316455019265

Graph of the ZZ-function along the critical line