↑

Properties

Label 20-2040e10-1.1-c1e10-0-1
Degree $20$
Conductor $1.248\times 10^{33}$
Sign $1$
Analytic cond. $1.31544\times 10^{12}$
Root an. cond. $4.03602$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 5·9-s + 10·11-s + 6·19-s − 8·25-s + 38·29-s + 12·31-s + 10·41-s + 10·45-s + 33·49-s − 20·55-s + 20·59-s + 32·61-s + 12·79-s + 15·81-s − 28·89-s − 12·95-s − 50·99-s − 16·109-s + 19·121-s + 10·125-s + 127-s + 131-s + 137-s + 139-s − 76·145-s + 149-s + ⋯
L(s)  = 1  − 0.894·5-s − 5/3·9-s + 3.01·11-s + 1.37·19-s − 8/5·25-s + 7.05·29-s + 2.15·31-s + 1.56·41-s + 1.49·45-s + 33/7·49-s − 2.69·55-s + 2.60·59-s + 4.09·61-s + 1.35·79-s + 5/3·81-s − 2.96·89-s − 1.23·95-s − 5.02·99-s − 1.53·109-s + 1.72·121-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6.31·145-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{10} \cdot 5^{10} \cdot 17^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{10} \cdot 5^{10} \cdot 17^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{30} \cdot 3^{10} \cdot 5^{10} \cdot 17^{10}\)
Sign: $1$
Analytic conductor: \(1.31544\times 10^{12}\)
Root analytic conductor: \(4.03602\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{30} \cdot 3^{10} \cdot 5^{10} \cdot 17^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(41.34013454\)
\(L(\frac12)\) \(\approx\) \(41.34013454\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T^{2} )^{5} \)
5 \( 1 + 2 T + 12 T^{2} + 6 p T^{3} + 61 T^{4} + 204 T^{5} + 61 p T^{6} + 6 p^{3} T^{7} + 12 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
17 \( ( 1 + T^{2} )^{5} \)
good7 \( 1 - 33 T^{2} + 485 T^{4} - 3992 T^{6} + 20378 T^{8} - 101678 T^{10} + 20378 p^{2} T^{12} - 3992 p^{4} T^{14} + 485 p^{6} T^{16} - 33 p^{8} T^{18} + p^{10} T^{20} \)
11 \( ( 1 - 5 T + 28 T^{2} - 85 T^{3} + 503 T^{4} - 1564 T^{5} + 503 p T^{6} - 85 p^{2} T^{7} + 28 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
13 \( 1 - 4 T^{2} - 242 T^{4} - 4438 T^{6} + 40561 T^{8} + 994964 T^{10} + 40561 p^{2} T^{12} - 4438 p^{4} T^{14} - 242 p^{6} T^{16} - 4 p^{8} T^{18} + p^{10} T^{20} \)
19 \( ( 1 - 3 T + 10 T^{2} - 17 T^{3} - 115 T^{4} + 148 T^{5} - 115 p T^{6} - 17 p^{2} T^{7} + 10 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
23 \( 1 - 120 T^{2} + 7162 T^{4} - 295774 T^{6} + 9502889 T^{8} - 244193932 T^{10} + 9502889 p^{2} T^{12} - 295774 p^{4} T^{14} + 7162 p^{6} T^{16} - 120 p^{8} T^{18} + p^{10} T^{20} \)
29 \( ( 1 - 19 T + 235 T^{2} - 2022 T^{3} + 14208 T^{4} - 82182 T^{5} + 14208 p T^{6} - 2022 p^{2} T^{7} + 235 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
31 \( ( 1 - 6 T + 113 T^{2} - 604 T^{3} + 6320 T^{4} - 25708 T^{5} + 6320 p T^{6} - 604 p^{2} T^{7} + 113 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
37 \( 1 - 101 T^{2} + 5381 T^{4} - 255544 T^{6} + 10525122 T^{8} - 378104390 T^{10} + 10525122 p^{2} T^{12} - 255544 p^{4} T^{14} + 5381 p^{6} T^{16} - 101 p^{8} T^{18} + p^{10} T^{20} \)
41 \( ( 1 - 5 T + 114 T^{2} - 345 T^{3} + 7105 T^{4} - 19092 T^{5} + 7105 p T^{6} - 345 p^{2} T^{7} + 114 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
43 \( 1 - 156 T^{2} + 17350 T^{4} - 1315702 T^{6} + 80494825 T^{8} - 3802710684 T^{10} + 80494825 p^{2} T^{12} - 1315702 p^{4} T^{14} + 17350 p^{6} T^{16} - 156 p^{8} T^{18} + p^{10} T^{20} \)
47 \( 1 - 205 T^{2} + 19021 T^{4} - 1228140 T^{6} + 71854930 T^{8} - 3736046094 T^{10} + 71854930 p^{2} T^{12} - 1228140 p^{4} T^{14} + 19021 p^{6} T^{16} - 205 p^{8} T^{18} + p^{10} T^{20} \)
53 \( 1 - 257 T^{2} + 32205 T^{4} - 2696796 T^{6} + 175296170 T^{8} - 9800030758 T^{10} + 175296170 p^{2} T^{12} - 2696796 p^{4} T^{14} + 32205 p^{6} T^{16} - 257 p^{8} T^{18} + p^{10} T^{20} \)
59 \( ( 1 - 10 T + 187 T^{2} - 1824 T^{3} + 20814 T^{4} - 137292 T^{5} + 20814 p T^{6} - 1824 p^{2} T^{7} + 187 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
61 \( ( 1 - 16 T + 351 T^{2} - 3736 T^{3} + 45452 T^{4} - 336976 T^{5} + 45452 p T^{6} - 3736 p^{2} T^{7} + 351 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
67 \( 1 - 418 T^{2} + 88549 T^{4} - 12393016 T^{6} + 1258333218 T^{8} - 96468609932 T^{10} + 1258333218 p^{2} T^{12} - 12393016 p^{4} T^{14} + 88549 p^{6} T^{16} - 418 p^{8} T^{18} + p^{10} T^{20} \)
71 \( ( 1 + 233 T^{2} - 736 T^{3} + 22992 T^{4} - 105408 T^{5} + 22992 p T^{6} - 736 p^{2} T^{7} + 233 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
73 \( 1 - 49 T^{2} + 16725 T^{4} - 986844 T^{6} + 137324858 T^{8} - 7723469478 T^{10} + 137324858 p^{2} T^{12} - 986844 p^{4} T^{14} + 16725 p^{6} T^{16} - 49 p^{8} T^{18} + p^{10} T^{20} \)
79 \( ( 1 - 6 T + 249 T^{2} - 2124 T^{3} + 28856 T^{4} - 258604 T^{5} + 28856 p T^{6} - 2124 p^{2} T^{7} + 249 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
83 \( 1 - 510 T^{2} + 132821 T^{4} - 22828776 T^{6} + 2851981842 T^{8} - 270185054708 T^{10} + 2851981842 p^{2} T^{12} - 22828776 p^{4} T^{14} + 132821 p^{6} T^{16} - 510 p^{8} T^{18} + p^{10} T^{20} \)
89 \( ( 1 + 14 T + 269 T^{2} + 2984 T^{3} + 38618 T^{4} + 373364 T^{5} + 38618 p T^{6} + 2984 p^{2} T^{7} + 269 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
97 \( 1 - 286 T^{2} + 46909 T^{4} - 5524104 T^{6} + 544996738 T^{8} - 50722521780 T^{10} + 544996738 p^{2} T^{12} - 5524104 p^{4} T^{14} + 46909 p^{6} T^{16} - 286 p^{8} T^{18} + p^{10} T^{20} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.22147234872229124359572385436, −3.02359316615942872417830956353, −2.91518552325028234938205267497, −2.89616451249139451563790612188, −2.80799925194872878861419111915, −2.79074106775423702656265219255, −2.61939803457332980767417589507, −2.59290843883328623097455113494, −2.39378196217654029623220437378, −2.36816710955269149703754758788, −2.29918440366176061391552495431, −2.15722075763661380454376524376, −2.07745712059852754253375549455, −1.68394579661448355688036053818, −1.60715870097864820627621164304, −1.57454830156971366383947409051, −1.25052343170836706069801541134, −1.20996984610073282303345985729, −1.14428823899972994717572051213, −0.950197023372952556806057348569, −0.893821648666570255505931884477, −0.62497041122395705868093134877, −0.61964879780348674921988618039, −0.58762354041983418426810430679, −0.35354394045370817964372968839, 0.35354394045370817964372968839, 0.58762354041983418426810430679, 0.61964879780348674921988618039, 0.62497041122395705868093134877, 0.893821648666570255505931884477, 0.950197023372952556806057348569, 1.14428823899972994717572051213, 1.20996984610073282303345985729, 1.25052343170836706069801541134, 1.57454830156971366383947409051, 1.60715870097864820627621164304, 1.68394579661448355688036053818, 2.07745712059852754253375549455, 2.15722075763661380454376524376, 2.29918440366176061391552495431, 2.36816710955269149703754758788, 2.39378196217654029623220437378, 2.59290843883328623097455113494, 2.61939803457332980767417589507, 2.79074106775423702656265219255, 2.80799925194872878861419111915, 2.89616451249139451563790612188, 2.91518552325028234938205267497, 3.02359316615942872417830956353, 3.22147234872229124359572385436

Graph of the $Z$-function along the critical line

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