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Properties

Label 8-546e4-1.1-c1e4-0-3
Degree 88
Conductor 8887314945688873149456
Sign 11
Analytic cond. 361.309361.309
Root an. cond. 2.088022.08802
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  − 2·3-s + 4-s + 9-s − 2·12-s + 6·17-s + 12·19-s + 8·23-s − 4·25-s + 2·27-s − 16·29-s + 36-s − 24·37-s − 24·41-s + 2·43-s + 49-s − 12·51-s − 40·53-s − 24·57-s + 6·59-s − 64-s + 18·67-s + 6·68-s − 16·69-s + 24·71-s + 8·75-s + 12·76-s − 24·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 1/3·9-s − 0.577·12-s + 1.45·17-s + 2.75·19-s + 1.66·23-s − 4/5·25-s + 0.384·27-s − 2.97·29-s + 1/6·36-s − 3.94·37-s − 3.74·41-s + 0.304·43-s + 1/7·49-s − 1.68·51-s − 5.49·53-s − 3.17·57-s + 0.781·59-s − 1/8·64-s + 2.19·67-s + 0.727·68-s − 1.92·69-s + 2.84·71-s + 0.923·75-s + 1.37·76-s − 2.70·79-s + ⋯

Functional equation

Λ(s)=((24⋅34⋅74⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((24⋅34⋅74⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \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: 24⋅34⋅74⋅1342^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}
Sign: 11
Analytic conductor: 361.309361.309
Root analytic conductor: 2.088022.08802
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅34⋅74⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.72153462440.7215346244
L(12)L(\frac12) ≈\approx 0.72153462440.7215346244
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)
bad2C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
3C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
7C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
13C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
good5C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
11C23C_2^3 1+6T2−85T4+6p2T6+p4T8 1 + 6 T^{2} - 85 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8}
17D4×C2D_4\times C_2 1−6T+5T2+18T3+60T4+18pT5+5p2T6−6p3T7+p4T8 1 - 6 T + 5 T^{2} + 18 T^{3} + 60 T^{4} + 18 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
19D4×C2D_4\times C_2 1−12T+82T2−408T3+1707T4−408pT5+82p2T6−12p3T7+p4T8 1 - 12 T + 82 T^{2} - 408 T^{3} + 1707 T^{4} - 408 p T^{5} + 82 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}
23D4×C2D_4\times C_2 1−8T+5T2−104T3+1480T4−104pT5+5p2T6−8p3T7+p4T8 1 - 8 T + 5 T^{2} - 104 T^{3} + 1480 T^{4} - 104 p T^{5} + 5 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
29C22C_2^2 (1+8T+35T2+8pT3+p2T4)2 ( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
31D4×C2D_4\times C_2 1−82T2+3171T4−82p2T6+p4T8 1 - 82 T^{2} + 3171 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8}
37D4×C2D_4\times C_2 1+24T+310T2+2832T3+19659T4+2832pT5+310p2T6+24p3T7+p4T8 1 + 24 T + 310 T^{2} + 2832 T^{3} + 19659 T^{4} + 2832 p T^{5} + 310 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}
41C22C_2^2 (1+12T+89T2+12pT3+p2T4)2 ( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1−2T−71T2+22T3+3604T4+22pT5−71p2T6−2p3T7+p4T8 1 - 2 T - 71 T^{2} + 22 T^{3} + 3604 T^{4} + 22 p T^{5} - 71 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
47D4×C2D_4\times C_2 1−132T2+8006T4−132p2T6+p4T8 1 - 132 T^{2} + 8006 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8}
53D4D_{4} (1+20T+203T2+20pT3+p2T4)2 ( 1 + 20 T + 203 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1−6T+69T2−342T3+476T4−342pT5+69p2T6−6p3T7+p4T8 1 - 6 T + 69 T^{2} - 342 T^{3} + 476 T^{4} - 342 p T^{5} + 69 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
61C23C_2^3 1−95T2+5304T4−95p2T6+p4T8 1 - 95 T^{2} + 5304 T^{4} - 95 p^{2} T^{6} + p^{4} T^{8}
67D4×C2D_4\times C_2 1−18T+265T2−2826T3+27396T4−2826pT5+265p2T6−18p3T7+p4T8 1 - 18 T + 265 T^{2} - 2826 T^{3} + 27396 T^{4} - 2826 p T^{5} + 265 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}
71D4×C2D_4\times C_2 1−24T+357T2−3960T3+35816T4−3960pT5+357p2T6−24p3T7+p4T8 1 - 24 T + 357 T^{2} - 3960 T^{3} + 35816 T^{4} - 3960 p T^{5} + 357 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}
73D4×C2D_4\times C_2 1−164T2+14310T4−164p2T6+p4T8 1 - 164 T^{2} + 14310 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8}
79D4D_{4} (1+12T+182T2+12pT3+p2T4)2 ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
83C22C_2^2 (1−163T2+p2T4)2 ( 1 - 163 T^{2} + p^{2} T^{4} )^{2}
89D4×C2D_4\times C_2 1+6T+93T2+486T3−292T4+486pT5+93p2T6+6p3T7+p4T8 1 + 6 T + 93 T^{2} + 486 T^{3} - 292 T^{4} + 486 p T^{5} + 93 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
97D4×C2D_4\times C_2 1−36T+730T2−10728T3+121299T4−10728pT5+730p2T6−36p3T7+p4T8 1 - 36 T + 730 T^{2} - 10728 T^{3} + 121299 T^{4} - 10728 p T^{5} + 730 p^{2} T^{6} - 36 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

−7.69142391746753347386998246056, −7.56608541158320025189730485218, −7.09334399883042678238062887757, −6.93224731100799593607005033012, −6.91557202736186785793798584034, −6.89939411990615936289009071744, −6.29043475376490839500308371656, −5.95253303556012599295410803858, −5.89574126155553846202212122519, −5.34705187045503085558831304047, −5.33227071454830488358601285270, −5.26073625750644198636961068661, −4.95476999836256925769902203774, −4.88049162606403145839739213054, −4.31654837945409076956688125335, −3.61383730384312203796128503229, −3.58100304210979593434574221023, −3.40070455090560381716613569849, −3.14827916740683880830491384810, −3.04019460697396748757509041738, −2.07883979412458789837696819307, −1.90383509974229487257351589042, −1.45735749955846818859094979571, −1.26226376673888705563999383917, −0.29548496073519443075110509941, 0.29548496073519443075110509941, 1.26226376673888705563999383917, 1.45735749955846818859094979571, 1.90383509974229487257351589042, 2.07883979412458789837696819307, 3.04019460697396748757509041738, 3.14827916740683880830491384810, 3.40070455090560381716613569849, 3.58100304210979593434574221023, 3.61383730384312203796128503229, 4.31654837945409076956688125335, 4.88049162606403145839739213054, 4.95476999836256925769902203774, 5.26073625750644198636961068661, 5.33227071454830488358601285270, 5.34705187045503085558831304047, 5.89574126155553846202212122519, 5.95253303556012599295410803858, 6.29043475376490839500308371656, 6.89939411990615936289009071744, 6.91557202736186785793798584034, 6.93224731100799593607005033012, 7.09334399883042678238062887757, 7.56608541158320025189730485218, 7.69142391746753347386998246056

Graph of the ZZ-function along the critical line