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Properties

Label 8-1920e4-1.1-c1e4-0-15
Degree 88
Conductor 135895.450×108135895.450\times 10^{8}
Sign 11
Analytic cond. 55247.555247.5
Root an. cond. 3.915513.91551
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  − 8·11-s − 8·13-s − 8·23-s − 2·25-s − 24·37-s − 24·47-s + 8·49-s − 24·59-s + 16·61-s + 16·71-s + 24·73-s − 9·81-s + 56·97-s + 32·107-s − 32·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 64·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + ⋯
L(s)  = 1  − 2.41·11-s − 2.21·13-s − 1.66·23-s − 2/5·25-s − 3.94·37-s − 3.50·47-s + 8/7·49-s − 3.12·59-s + 2.04·61-s + 1.89·71-s + 2.80·73-s − 81-s + 5.68·97-s + 3.09·107-s − 3.06·109-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.35·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.76·169-s + ⋯

Functional equation

Λ(s)=((228⋅34⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((228⋅34⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 228⋅34⋅542^{28} \cdot 3^{4} \cdot 5^{4}
Sign: 11
Analytic conductor: 55247.555247.5
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 228⋅34⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{28} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.48528490870.4852849087
L(12)L(\frac12) ≈\approx 0.48528490870.4852849087
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)
bad2 1 1
3C22C_2^2 1+p2T4 1 + p^{2} T^{4}
5C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
good7D4×C2D_4\times C_2 1−8T2+18T4−8p2T6+p4T8 1 - 8 T^{2} + 18 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8}
11C2C_2 (1+2T+pT2)4 ( 1 + 2 T + p T^{2} )^{4}
13D4D_{4} (1+4T+6T2+4pT3+p2T4)2 ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
17C2C_2 (1−8T+pT2)2(1+8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2}
19C22C_2^2 (1−34T2+p2T4)2 ( 1 - 34 T^{2} + p^{2} T^{4} )^{2}
23D4D_{4} (1+4T+44T2+4pT3+p2T4)2 ( 1 + 4 T + 44 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
29D4×C2D_4\times C_2 1−36T2+470T4−36p2T6+p4T8 1 - 36 T^{2} + 470 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8}
31C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
37C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
41D4×C2D_4\times C_2 1+36T2+2150T4+36p2T6+p4T8 1 + 36 T^{2} + 2150 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8}
43D4×C2D_4\times C_2 1−128T2+7410T4−128p2T6+p4T8 1 - 128 T^{2} + 7410 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8}
47D4D_{4} (1+12T+124T2+12pT3+p2T4)2 ( 1 + 12 T + 124 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
53D4×C2D_4\times C_2 1−156T2+11318T4−156p2T6+p4T8 1 - 156 T^{2} + 11318 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1+12T+130T2+12pT3+p2T4)2 ( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
61D4D_{4} (1−8T+114T2−8pT3+p2T4)2 ( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−128T2+9618T4−128p2T6+p4T8 1 - 128 T^{2} + 9618 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1−8T+134T2−8pT3+p2T4)2 ( 1 - 8 T + 134 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
73D4D_{4} (1−12T+158T2−12pT3+p2T4)2 ( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
79C22C_2^2 (1−134T2+p2T4)2 ( 1 - 134 T^{2} + p^{2} T^{4} )^{2}
83C22C_2^2 (1+16T2+p2T4)2 ( 1 + 16 T^{2} + p^{2} T^{4} )^{2}
89C22C_2^2 (1−82T2+p2T4)2 ( 1 - 82 T^{2} + p^{2} T^{4} )^{2}
97D4D_{4} (1−28T+366T2−28pT3+p2T4)2 ( 1 - 28 T + 366 T^{2} - 28 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

−6.59655441678853838811752721281, −6.42896644558877658119702306624, −6.17984692714758987043493697691, −5.82356876307410250197817252203, −5.53378606433946292123664802819, −5.31956343266321610278853432980, −5.31363040734891891155481386316, −5.17865964691239363458411811176, −4.77819087429970326287697439145, −4.69143699562721448389029417112, −4.65868721704601283607673988320, −4.04937795802564718577397406734, −4.03149006555780361561473303728, −3.47977715582030238510887554757, −3.40191639350684731660776267582, −3.17333351820532575230046618061, −3.04149343987728708260196947300, −2.63225489228734166587794567602, −2.12836691633548059577567627758, −2.12319551884840678185293684285, −1.93503638294745180975540520550, −1.88814021776235219400720709916, −1.09943476519666678636660774401, −0.40919772170541746698573003016, −0.22801809927001565572479610646, 0.22801809927001565572479610646, 0.40919772170541746698573003016, 1.09943476519666678636660774401, 1.88814021776235219400720709916, 1.93503638294745180975540520550, 2.12319551884840678185293684285, 2.12836691633548059577567627758, 2.63225489228734166587794567602, 3.04149343987728708260196947300, 3.17333351820532575230046618061, 3.40191639350684731660776267582, 3.47977715582030238510887554757, 4.03149006555780361561473303728, 4.04937795802564718577397406734, 4.65868721704601283607673988320, 4.69143699562721448389029417112, 4.77819087429970326287697439145, 5.17865964691239363458411811176, 5.31363040734891891155481386316, 5.31956343266321610278853432980, 5.53378606433946292123664802819, 5.82356876307410250197817252203, 6.17984692714758987043493697691, 6.42896644558877658119702306624, 6.59655441678853838811752721281

Graph of the ZZ-function along the critical line