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Properties

Label 8-1575e4-1.1-c3e4-0-7
Degree 88
Conductor 6.154×10126.154\times 10^{12}
Sign 11
Analytic cond. 7.45738×1077.45738\times 10^{7}
Root an. cond. 9.639919.63991
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 11·4-s + 28·7-s − 22·13-s + 45·16-s − 132·19-s − 308·28-s − 126·31-s − 354·37-s − 272·43-s + 490·49-s + 242·52-s − 1.17e3·61-s − 363·64-s − 38·67-s − 188·73-s + 1.45e3·76-s − 690·79-s − 616·91-s + 1.98e3·97-s − 1.98e3·103-s − 2.55e3·109-s + 1.26e3·112-s − 4.75e3·121-s + 1.38e3·124-s + 127-s + 131-s − 3.69e3·133-s + ⋯
L(s)  = 1  − 1.37·4-s + 1.51·7-s − 0.469·13-s + 0.703·16-s − 1.59·19-s − 2.07·28-s − 0.730·31-s − 1.57·37-s − 0.964·43-s + 10/7·49-s + 0.645·52-s − 2.45·61-s − 0.708·64-s − 0.0692·67-s − 0.301·73-s + 2.19·76-s − 0.982·79-s − 0.709·91-s + 2.07·97-s − 1.89·103-s − 2.24·109-s + 1.06·112-s − 3.57·121-s + 1.00·124-s + 0.000698·127-s + 0.000666·131-s − 2.40·133-s + ⋯

Functional equation

Λ(s)=((38⋅58⋅74)s/2 ΓC(s)4 L(s)=( Λ(4−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(4-s)\end{aligned}
Λ(s)=((38⋅58⋅74)s/2 ΓC(s+3/2)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+3/2)^{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: 7.45738×1077.45738\times 10^{7}
Root analytic conductor: 9.639919.63991
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 38⋅58⋅74, ( :3/2,3/2,3/2,3/2), 1)(8,\ 3^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
7C1C_1 (1−pT)4 ( 1 - p T )^{4}
good2C22≀C2C_2^2 \wr C_2 1+11T2+19p2T4+11p6T6+p12T8 1 + 11 T^{2} + 19 p^{2} T^{4} + 11 p^{6} T^{6} + p^{12} T^{8}
11C22≀C2C_2^2 \wr C_2 1+4757T2+9131212T4+4757p6T6+p12T8 1 + 4757 T^{2} + 9131212 T^{4} + 4757 p^{6} T^{6} + p^{12} T^{8}
13D4D_{4} (1+11T+334pT2+11p3T3+p6T4)2 ( 1 + 11 T + 334 p T^{2} + 11 p^{3} T^{3} + p^{6} T^{4} )^{2}
17C22≀C2C_2^2 \wr C_2 1+16103T2+110752648T4+16103p6T6+p12T8 1 + 16103 T^{2} + 110752648 T^{4} + 16103 p^{6} T^{6} + p^{12} T^{8}
19D4D_{4} (1+66T+762pT2+66p3T3+p6T4)2 ( 1 + 66 T + 762 p T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} )^{2}
23C22≀C2C_2^2 \wr C_2 1+16580T2+227073517T4+16580p6T6+p12T8 1 + 16580 T^{2} + 227073517 T^{4} + 16580 p^{6} T^{6} + p^{12} T^{8}
29C22≀C2C_2^2 \wr C_2 1+57852T2+1698017789T4+57852p6T6+p12T8 1 + 57852 T^{2} + 1698017789 T^{4} + 57852 p^{6} T^{6} + p^{12} T^{8}
31D4D_{4} (1+63T+42068T2+63p3T3+p6T4)2 ( 1 + 63 T + 42068 T^{2} + 63 p^{3} T^{3} + p^{6} T^{4} )^{2}
37D4D_{4} (1+177T+90632T2+177p3T3+p6T4)2 ( 1 + 177 T + 90632 T^{2} + 177 p^{3} T^{3} + p^{6} T^{4} )^{2}
41C22≀C2C_2^2 \wr C_2 1+93635T2+8952796516T4+93635p6T6+p12T8 1 + 93635 T^{2} + 8952796516 T^{4} + 93635 p^{6} T^{6} + p^{12} T^{8}
43D4D_{4} (1+136T+147517T2+136p3T3+p6T4)2 ( 1 + 136 T + 147517 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} )^{2}
47C22≀C2C_2^2 \wr C_2 1+174744T2+15146248718T4+174744p6T6+p12T8 1 + 174744 T^{2} + 15146248718 T^{4} + 174744 p^{6} T^{6} + p^{12} T^{8}
53C22≀C2C_2^2 \wr C_2 1+494267T2+102952583068T4+494267p6T6+p12T8 1 + 494267 T^{2} + 102952583068 T^{4} + 494267 p^{6} T^{6} + p^{12} T^{8}
59C22≀C2C_2^2 \wr C_2 1+646299T2+187556725712T4+646299p6T6+p12T8 1 + 646299 T^{2} + 187556725712 T^{4} + 646299 p^{6} T^{6} + p^{12} T^{8}
61D4D_{4} (1+585T+372962T2+585p3T3+p6T4)2 ( 1 + 585 T + 372962 T^{2} + 585 p^{3} T^{3} + p^{6} T^{4} )^{2}
67D4D_{4} (1+19T+404134T2+19p3T3+p6T4)2 ( 1 + 19 T + 404134 T^{2} + 19 p^{3} T^{3} + p^{6} T^{4} )^{2}
71C22≀C2C_2^2 \wr C_2 1+397485T2+173980338356T4+397485p6T6+p12T8 1 + 397485 T^{2} + 173980338356 T^{4} + 397485 p^{6} T^{6} + p^{12} T^{8}
73D4D_{4} (1+94T+606202T2+94p3T3+p6T4)2 ( 1 + 94 T + 606202 T^{2} + 94 p^{3} T^{3} + p^{6} T^{4} )^{2}
79D4D_{4} (1+345T+1001934T2+345p3T3+p6T4)2 ( 1 + 345 T + 1001934 T^{2} + 345 p^{3} T^{3} + p^{6} T^{4} )^{2}
83C22≀C2C_2^2 \wr C_2 1+1768119T2+1429880388068T4+1768119p6T6+p12T8 1 + 1768119 T^{2} + 1429880388068 T^{4} + 1768119 p^{6} T^{6} + p^{12} T^{8}
89C22≀C2C_2^2 \wr C_2 1+1398400T2+1454863323486T4+1398400p6T6+p12T8 1 + 1398400 T^{2} + 1454863323486 T^{4} + 1398400 p^{6} T^{6} + p^{12} T^{8}
97D4D_{4} (1−990T+1951602T2−990p3T3+p6T4)2 ( 1 - 990 T + 1951602 T^{2} - 990 p^{3} T^{3} + p^{6} 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

−6.76529487309728492491636262408, −6.59271941974846036949428184158, −6.21133145015693867588509372853, −6.05631413059383073384896999307, −6.00143164801777872697499944006, −5.55685221795902748955977923649, −5.23102834602827771244710121409, −5.22088089529499081455848141947, −5.17004850143888157817590030889, −4.56563497805090906697797444809, −4.51038209326343143389820487526, −4.49170702716514902735483591877, −4.43962317105888616865172870694, −3.73589721903544306901852113696, −3.64966226670344065188603638656, −3.64839936346230240244506448210, −3.27150044087074957163924660323, −2.68392326501282368897616330100, −2.65622382088595926026509903834, −2.31724247039840670114315912482, −1.95332170470552353019557658833, −1.84447864536553326190305873046, −1.25115933733093710893864531613, −1.19764643516750001531035071499, −1.13093303974104489748271046879, 0, 0, 0, 0, 1.13093303974104489748271046879, 1.19764643516750001531035071499, 1.25115933733093710893864531613, 1.84447864536553326190305873046, 1.95332170470552353019557658833, 2.31724247039840670114315912482, 2.65622382088595926026509903834, 2.68392326501282368897616330100, 3.27150044087074957163924660323, 3.64839936346230240244506448210, 3.64966226670344065188603638656, 3.73589721903544306901852113696, 4.43962317105888616865172870694, 4.49170702716514902735483591877, 4.51038209326343143389820487526, 4.56563497805090906697797444809, 5.17004850143888157817590030889, 5.22088089529499081455848141947, 5.23102834602827771244710121409, 5.55685221795902748955977923649, 6.00143164801777872697499944006, 6.05631413059383073384896999307, 6.21133145015693867588509372853, 6.59271941974846036949428184158, 6.76529487309728492491636262408

Graph of the ZZ-function along the critical line