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Properties

Label 16-525e8-1.1-c0e8-0-0
Degree 1616
Conductor 5.771×10215.771\times 10^{21}
Sign 11
Analytic cond. 2.22091×10−52.22091\times 10^{-5}
Root an. cond. 0.5118680.511868
Motivic weight 00
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  + 2·16-s − 12·61-s + 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯
L(s)  = 1  + 2·16-s − 12·61-s + 81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯

Functional equation

Λ(s)=((38⋅516⋅78)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((38⋅516⋅78)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 38⋅516⋅783^{8} \cdot 5^{16} \cdot 7^{8}
Sign: 11
Analytic conductor: 2.22091×10−52.22091\times 10^{-5}
Root analytic conductor: 0.5118680.511868
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 38⋅516⋅78, ( :[0]8), 1)(16,\ 3^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.34066150680.3406615068
L(12)L(\frac12) ≈\approx 0.34066150680.3406615068
L(1)L(1) 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}
ppFp(T)F_p(T)
bad3 1−T4+T8 1 - T^{4} + T^{8}
5 1 1
7 1−T4+T8 1 - T^{4} + T^{8}
good2 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
11 (1−T+T2)4(1+T+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}
13 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
17 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
19 (1+T2)4(1−T2+T4)2 ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}
23 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
29 (1+T2)8 ( 1 + T^{2} )^{8}
31 (1−T+T2)4(1+T+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}
37 (1+T4)2(1−T4+T8) ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )
41 (1+T2)8 ( 1 + T^{2} )^{8}
43 (1+T4)4 ( 1 + T^{4} )^{4}
47 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
53 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
59 (1−T+T2)4(1+T+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}
61 (1+T)8(1+T+T2)4 ( 1 + T )^{8}( 1 + T + T^{2} )^{4}
67 (1+T4)2(1−T4+T8) ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )
71 (1−T)8(1+T)8 ( 1 - T )^{8}( 1 + T )^{8}
73 (1+T4)2(1−T4+T8) ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )
79 (1+T2)4(1−T2+T4)2 ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}
83 (1+T4)4 ( 1 + T^{4} )^{4}
89 (1−T+T2)4(1+T+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}
97 (1−T4+T8)2 ( 1 - T^{4} + T^{8} )^{2}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.14959005626572939246939861449, −4.71787999659229431119451120672, −4.71368681971134051877329674142, −4.50364774584328870421161627358, −4.50189451263717730946963683119, −4.42602831652703613530353595059, −4.37837114333564791038553300389, −4.07759666830518613715473251466, −3.91490788644184588606427867294, −3.72425329598832982644511654373, −3.34047050755471509032231667467, −3.31966094801932222018327337308, −3.28412892766438771131336611196, −3.28069687518321590505497826757, −3.16921841480485411844525611973, −2.69936208651861485005943151331, −2.68599220391535101980008274751, −2.58214667023386001339226008871, −2.36390133880571560857729700542, −1.93422821101657829483960274435, −1.75017213250817479610146388580, −1.55080816345337616069209461415, −1.48726815656126762725697522187, −1.34065392843507600280010637163, −0.957285847337279441362749789493, 0.957285847337279441362749789493, 1.34065392843507600280010637163, 1.48726815656126762725697522187, 1.55080816345337616069209461415, 1.75017213250817479610146388580, 1.93422821101657829483960274435, 2.36390133880571560857729700542, 2.58214667023386001339226008871, 2.68599220391535101980008274751, 2.69936208651861485005943151331, 3.16921841480485411844525611973, 3.28069687518321590505497826757, 3.28412892766438771131336611196, 3.31966094801932222018327337308, 3.34047050755471509032231667467, 3.72425329598832982644511654373, 3.91490788644184588606427867294, 4.07759666830518613715473251466, 4.37837114333564791038553300389, 4.42602831652703613530353595059, 4.50189451263717730946963683119, 4.50364774584328870421161627358, 4.71368681971134051877329674142, 4.71787999659229431119451120672, 5.14959005626572939246939861449

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.