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Properties

Label 20-1305e10-1.1-c1e10-0-5
Degree $20$
Conductor $1.433\times 10^{31}$
Sign $1$
Analytic cond. $1.50964\times 10^{10}$
Root an. cond. $3.22807$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·4-s + 10·5-s − 4·7-s + 4·13-s + 21·16-s + 70·20-s + 10·23-s + 55·25-s − 28·28-s − 40·35-s − 32·49-s + 28·52-s − 14·53-s + 32·59-s + 38·64-s + 40·65-s − 8·67-s − 8·71-s + 210·80-s + 18·83-s − 16·91-s + 70·92-s + 385·100-s − 12·103-s + 32·107-s − 26·109-s − 84·112-s + ⋯
L(s)  = 1  + 7/2·4-s + 4.47·5-s − 1.51·7-s + 1.10·13-s + 21/4·16-s + 15.6·20-s + 2.08·23-s + 11·25-s − 5.29·28-s − 6.76·35-s − 4.57·49-s + 3.88·52-s − 1.92·53-s + 4.16·59-s + 19/4·64-s + 4.96·65-s − 0.977·67-s − 0.949·71-s + 23.4·80-s + 1.97·83-s − 1.67·91-s + 7.29·92-s + 77/2·100-s − 1.18·103-s + 3.09·107-s − 2.49·109-s − 7.93·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(3^{20} \cdot 5^{10} \cdot 29^{10}\)
Sign: $1$
Analytic conductor: \(1.50964\times 10^{10}\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 3^{20} \cdot 5^{10} \cdot 29^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(80.81670571\)
\(L(\frac12)\) \(\approx\) \(80.81670571\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 - T )^{10} \)
29 \( 1 - 43 T^{2} + 144 T^{3} + 694 T^{4} - 5472 T^{5} + 694 p T^{6} + 144 p^{2} T^{7} - 43 p^{3} T^{8} + p^{5} T^{10} \)
good2 \( 1 - 7 T^{2} + 7 p^{2} T^{4} - 87 T^{6} + 225 T^{8} - 61 p^{3} T^{10} + 225 p^{2} T^{12} - 87 p^{4} T^{14} + 7 p^{8} T^{16} - 7 p^{8} T^{18} + p^{10} T^{20} \)
7 \( ( 1 + 2 T + 22 T^{2} + 48 T^{3} + 255 T^{4} + 460 T^{5} + 255 p T^{6} + 48 p^{2} T^{7} + 22 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
11 \( 1 - 63 T^{2} + 2088 T^{4} - 4219 p T^{6} + 68969 p T^{8} - 9488388 T^{10} + 68969 p^{3} T^{12} - 4219 p^{5} T^{14} + 2088 p^{6} T^{16} - 63 p^{8} T^{18} + p^{10} T^{20} \)
13 \( ( 1 - 2 T + 40 T^{2} - 94 T^{3} + 743 T^{4} - 1760 T^{5} + 743 p T^{6} - 94 p^{2} T^{7} + 40 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
17 \( 1 - 108 T^{2} + 5754 T^{4} - 199502 T^{6} + 5024917 T^{8} - 96868908 T^{10} + 5024917 p^{2} T^{12} - 199502 p^{4} T^{14} + 5754 p^{6} T^{16} - 108 p^{8} T^{18} + p^{10} T^{20} \)
19 \( 1 - 50 T^{2} + 1561 T^{4} - 47584 T^{6} + 1147118 T^{8} - 22382108 T^{10} + 1147118 p^{2} T^{12} - 47584 p^{4} T^{14} + 1561 p^{6} T^{16} - 50 p^{8} T^{18} + p^{10} T^{20} \)
23 \( ( 1 - 5 T + 29 T^{2} - 190 T^{3} + 1084 T^{4} - 2802 T^{5} + 1084 p T^{6} - 190 p^{2} T^{7} + 29 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
31 \( 1 - 98 T^{2} + 6529 T^{4} - 318400 T^{6} + 12468542 T^{8} - 419625404 T^{10} + 12468542 p^{2} T^{12} - 318400 p^{4} T^{14} + 6529 p^{6} T^{16} - 98 p^{8} T^{18} + p^{10} T^{20} \)
37 \( 1 - 141 T^{2} + 10701 T^{4} - 602732 T^{6} + 28260106 T^{8} - 1135031022 T^{10} + 28260106 p^{2} T^{12} - 602732 p^{4} T^{14} + 10701 p^{6} T^{16} - 141 p^{8} T^{18} + p^{10} T^{20} \)
41 \( 1 - 213 T^{2} + 20109 T^{4} - 1191404 T^{6} + 55446274 T^{8} - 2340896190 T^{10} + 55446274 p^{2} T^{12} - 1191404 p^{4} T^{14} + 20109 p^{6} T^{16} - 213 p^{8} T^{18} + p^{10} T^{20} \)
43 \( 1 - 77 T^{2} + 4009 T^{4} - 128788 T^{6} + 4747502 T^{8} - 121298462 T^{10} + 4747502 p^{2} T^{12} - 128788 p^{4} T^{14} + 4009 p^{6} T^{16} - 77 p^{8} T^{18} + p^{10} T^{20} \)
47 \( 1 - 156 T^{2} + 14778 T^{4} - 1081550 T^{6} + 66931429 T^{8} - 3454344300 T^{10} + 66931429 p^{2} T^{12} - 1081550 p^{4} T^{14} + 14778 p^{6} T^{16} - 156 p^{8} T^{18} + p^{10} T^{20} \)
53 \( ( 1 + 7 T + 131 T^{2} + 692 T^{3} + 7864 T^{4} + 40506 T^{5} + 7864 p T^{6} + 692 p^{2} T^{7} + 131 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
59 \( ( 1 - 16 T + 207 T^{2} - 1832 T^{3} + 326 p T^{4} - 151440 T^{5} + 326 p^{2} T^{6} - 1832 p^{2} T^{7} + 207 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
61 \( 1 - 98 T^{2} + 5845 T^{4} - 314776 T^{6} + 4040882 T^{8} + 295409332 T^{10} + 4040882 p^{2} T^{12} - 314776 p^{4} T^{14} + 5845 p^{6} T^{16} - 98 p^{8} T^{18} + p^{10} T^{20} \)
67 \( ( 1 + 4 T + 114 T^{2} - 406 T^{3} - 493 T^{4} - 87480 T^{5} - 493 p T^{6} - 406 p^{2} T^{7} + 114 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
71 \( ( 1 + 4 T + 123 T^{2} - 520 T^{3} - 734 T^{4} - 110712 T^{5} - 734 p T^{6} - 520 p^{2} T^{7} + 123 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
73 \( 1 - 253 T^{2} + 39645 T^{4} - 4613484 T^{6} + 438159090 T^{8} - 34455047022 T^{10} + 438159090 p^{2} T^{12} - 4613484 p^{4} T^{14} + 39645 p^{6} T^{16} - 253 p^{8} T^{18} + p^{10} T^{20} \)
79 \( 1 - 650 T^{2} + 198961 T^{4} - 37714144 T^{6} + 4896455918 T^{8} - 454601975468 T^{10} + 4896455918 p^{2} T^{12} - 37714144 p^{4} T^{14} + 198961 p^{6} T^{16} - 650 p^{8} T^{18} + p^{10} T^{20} \)
83 \( ( 1 - 9 T + 263 T^{2} - 1782 T^{3} + 36982 T^{4} - 210474 T^{5} + 36982 p T^{6} - 1782 p^{2} T^{7} + 263 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
89 \( 1 - 384 T^{2} + 67914 T^{4} - 7684778 T^{6} + 697282165 T^{8} - 60978237420 T^{10} + 697282165 p^{2} T^{12} - 7684778 p^{4} T^{14} + 67914 p^{6} T^{16} - 384 p^{8} T^{18} + p^{10} T^{20} \)
97 \( 1 - 537 T^{2} + 153441 T^{4} - 29755892 T^{6} + 4266796702 T^{8} - 469798229862 T^{10} + 4266796702 p^{2} T^{12} - 29755892 p^{4} T^{14} + 153441 p^{6} T^{16} - 537 p^{8} T^{18} + p^{10} T^{20} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.46179980494234207438018244444, −3.27623997674716193171561456092, −3.05913825159591590495255191075, −3.05353602903702437527152719439, −2.95675921724181644586067679350, −2.85036291793179715963463685917, −2.80180642836714844777338434005, −2.74596325215710833828610351377, −2.42706393255930914624360446320, −2.42018611861617849638536202071, −2.39432256676549457462502040460, −2.31249164834196704487223249013, −2.04858476779676954544920378846, −2.01348695212329051493825833547, −1.84045972043443065607120118420, −1.80968549024770878685810068534, −1.63157727610777329530468133378, −1.61017742538006411323771099594, −1.47709398145327119030735771265, −1.26346530003659807665397936750, −1.18446826352011163353187655556, −1.00057809024180993505295985606, −0.70768930642772335663164903114, −0.64516462390349420533659826536, −0.25642613781027576695368636259, 0.25642613781027576695368636259, 0.64516462390349420533659826536, 0.70768930642772335663164903114, 1.00057809024180993505295985606, 1.18446826352011163353187655556, 1.26346530003659807665397936750, 1.47709398145327119030735771265, 1.61017742538006411323771099594, 1.63157727610777329530468133378, 1.80968549024770878685810068534, 1.84045972043443065607120118420, 2.01348695212329051493825833547, 2.04858476779676954544920378846, 2.31249164834196704487223249013, 2.39432256676549457462502040460, 2.42018611861617849638536202071, 2.42706393255930914624360446320, 2.74596325215710833828610351377, 2.80180642836714844777338434005, 2.85036291793179715963463685917, 2.95675921724181644586067679350, 3.05353602903702437527152719439, 3.05913825159591590495255191075, 3.27623997674716193171561456092, 3.46179980494234207438018244444

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

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