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Properties

Label 20-9280e10-1.1-c1e10-0-1
Degree $20$
Conductor $4.737\times 10^{39}$
Sign $1$
Analytic cond. $4.99172\times 10^{18}$
Root an. cond. $8.60820$
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  + 10·5-s − 7·9-s − 6·13-s + 18·17-s + 55·25-s + 10·29-s + 4·37-s − 70·45-s − 23·49-s + 14·53-s − 6·61-s − 60·65-s + 42·73-s + 27·81-s + 180·85-s + 8·89-s + 62·97-s + 14·101-s − 24·109-s + 62·113-s + 42·117-s − 42·121-s + 220·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 4.47·5-s − 7/3·9-s − 1.66·13-s + 4.36·17-s + 11·25-s + 1.85·29-s + 0.657·37-s − 10.4·45-s − 3.28·49-s + 1.92·53-s − 0.768·61-s − 7.44·65-s + 4.91·73-s + 3·81-s + 19.5·85-s + 0.847·89-s + 6.29·97-s + 1.39·101-s − 2.29·109-s + 5.83·113-s + 3.88·117-s − 3.81·121-s + 19.6·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 5^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 5^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{60} \cdot 5^{10} \cdot 29^{10}\)
Sign: $1$
Analytic conductor: \(4.99172\times 10^{18}\)
Root analytic conductor: \(8.60820\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{60} \cdot 5^{10} \cdot 29^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(132.7506095\)
\(L(\frac12)\) \(\approx\) \(132.7506095\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 - T )^{10} \)
29 \( ( 1 - T )^{10} \)
good3 \( 1 + 7 T^{2} + 22 T^{4} + 46 T^{6} + 77 T^{8} + 110 T^{10} + 77 p^{2} T^{12} + 46 p^{4} T^{14} + 22 p^{6} T^{16} + 7 p^{8} T^{18} + p^{10} T^{20} \)
7 \( 1 + 23 T^{2} + 314 T^{4} + 3394 T^{6} + 29497 T^{8} + 217622 T^{10} + 29497 p^{2} T^{12} + 3394 p^{4} T^{14} + 314 p^{6} T^{16} + 23 p^{8} T^{18} + p^{10} T^{20} \)
11 \( 1 + 42 T^{2} + 921 T^{4} + 15456 T^{6} + 218302 T^{8} + 2605996 T^{10} + 218302 p^{2} T^{12} + 15456 p^{4} T^{14} + 921 p^{6} T^{16} + 42 p^{8} T^{18} + p^{10} T^{20} \)
13 \( ( 1 + 3 T + 24 T^{2} + 6 p T^{3} + 459 T^{4} + 1462 T^{5} + 459 p T^{6} + 6 p^{3} T^{7} + 24 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
17 \( ( 1 - 9 T + 52 T^{2} - 366 T^{3} + 1885 T^{4} - 7406 T^{5} + 1885 p T^{6} - 366 p^{2} T^{7} + 52 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
19 \( 1 + 98 T^{2} + 4457 T^{4} + 129088 T^{6} + 2858782 T^{8} + 55961852 T^{10} + 2858782 p^{2} T^{12} + 129088 p^{4} T^{14} + 4457 p^{6} T^{16} + 98 p^{8} T^{18} + p^{10} T^{20} \)
23 \( 1 + 83 T^{2} + 3650 T^{4} + 126186 T^{6} + 3760721 T^{8} + 94417422 T^{10} + 3760721 p^{2} T^{12} + 126186 p^{4} T^{14} + 3650 p^{6} T^{16} + 83 p^{8} T^{18} + p^{10} T^{20} \)
31 \( 1 + 107 T^{2} + 8242 T^{4} + 453930 T^{6} + 19294401 T^{8} + 674427230 T^{10} + 19294401 p^{2} T^{12} + 453930 p^{4} T^{14} + 8242 p^{6} T^{16} + 107 p^{8} T^{18} + p^{10} T^{20} \)
37 \( ( 1 - 2 T + 119 T^{2} - 192 T^{3} + 7228 T^{4} - 10380 T^{5} + 7228 p T^{6} - 192 p^{2} T^{7} + 119 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
41 \( ( 1 + 109 T^{2} + 280 T^{3} + 6186 T^{4} + 18928 T^{5} + 6186 p T^{6} + 280 p^{2} T^{7} + 109 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
43 \( 1 + 179 T^{2} + 20022 T^{4} + 1520990 T^{6} + 91166765 T^{8} + 4308802806 T^{10} + 91166765 p^{2} T^{12} + 1520990 p^{4} T^{14} + 20022 p^{6} T^{16} + 179 p^{8} T^{18} + p^{10} T^{20} \)
47 \( 1 + 202 T^{2} + 24993 T^{4} + 2179200 T^{6} + 145004478 T^{8} + 7654790636 T^{10} + 145004478 p^{2} T^{12} + 2179200 p^{4} T^{14} + 24993 p^{6} T^{16} + 202 p^{8} T^{18} + p^{10} T^{20} \)
53 \( ( 1 - 7 T + 152 T^{2} - 706 T^{3} + 8907 T^{4} - 34998 T^{5} + 8907 p T^{6} - 706 p^{2} T^{7} + 152 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
59 \( 1 + 183 T^{2} + 27166 T^{4} + 2639846 T^{6} + 220664769 T^{8} + 13948523334 T^{10} + 220664769 p^{2} T^{12} + 2639846 p^{4} T^{14} + 27166 p^{6} T^{16} + 183 p^{8} T^{18} + p^{10} T^{20} \)
61 \( ( 1 + 3 T + 182 T^{2} + 1242 T^{3} + 14125 T^{4} + 129166 T^{5} + 14125 p T^{6} + 1242 p^{2} T^{7} + 182 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
67 \( 1 + 350 T^{2} + 54905 T^{4} + 5049632 T^{6} + 321429870 T^{8} + 19339951428 T^{10} + 321429870 p^{2} T^{12} + 5049632 p^{4} T^{14} + 54905 p^{6} T^{16} + 350 p^{8} T^{18} + p^{10} T^{20} \)
71 \( 1 + 374 T^{2} + 74093 T^{4} + 10209576 T^{6} + 1054267970 T^{8} + 84396281668 T^{10} + 1054267970 p^{2} T^{12} + 10209576 p^{4} T^{14} + 74093 p^{6} T^{16} + 374 p^{8} T^{18} + p^{10} T^{20} \)
73 \( ( 1 - 21 T + 208 T^{2} - 1702 T^{3} + 22161 T^{4} - 241950 T^{5} + 22161 p T^{6} - 1702 p^{2} T^{7} + 208 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
79 \( 1 + 575 T^{2} + 159562 T^{4} + 28186434 T^{6} + 3508956681 T^{8} + 321290071110 T^{10} + 3508956681 p^{2} T^{12} + 28186434 p^{4} T^{14} + 159562 p^{6} T^{16} + 575 p^{8} T^{18} + p^{10} T^{20} \)
83 \( 1 + 250 T^{2} + 43465 T^{4} + 5021280 T^{6} + 494553918 T^{8} + 41740876428 T^{10} + 494553918 p^{2} T^{12} + 5021280 p^{4} T^{14} + 43465 p^{6} T^{16} + 250 p^{8} T^{18} + p^{10} T^{20} \)
89 \( ( 1 - 4 T + 265 T^{2} - 1480 T^{3} + 33710 T^{4} - 196328 T^{5} + 33710 p T^{6} - 1480 p^{2} T^{7} + 265 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
97 \( ( 1 - 31 T + 732 T^{2} - 12026 T^{3} + 160637 T^{4} - 1739234 T^{5} + 160637 p T^{6} - 12026 p^{2} T^{7} + 732 p^{3} T^{8} - 31 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
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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

−2.52350246997799983206286316560, −2.44023882643598481388493392736, −2.42740358549198249010483388323, −2.41469179236214248949740474852, −2.37659138727350784648141047916, −2.14136880642560356831141629808, −1.99130131160294896058880834494, −1.98346944974395038378352768445, −1.84363909086548750036146894973, −1.79111576736196638703528689575, −1.77945837814196741939444883086, −1.63706000214828621402987371155, −1.54098411627574639238341453522, −1.46872167963424803085934063680, −1.44606671930812037507367017359, −1.04780034467168008844133484084, −1.04472740699508266838861258291, −1.01141507522903470635733248117, −0.853007838535907261393565615454, −0.77084948887440100124896152278, −0.72456822963847268613342069547, −0.55897822654879361351187676629, −0.55278826151689006021304805070, −0.34197475031547702478520241450, −0.17937614966760350156148122395, 0.17937614966760350156148122395, 0.34197475031547702478520241450, 0.55278826151689006021304805070, 0.55897822654879361351187676629, 0.72456822963847268613342069547, 0.77084948887440100124896152278, 0.853007838535907261393565615454, 1.01141507522903470635733248117, 1.04472740699508266838861258291, 1.04780034467168008844133484084, 1.44606671930812037507367017359, 1.46872167963424803085934063680, 1.54098411627574639238341453522, 1.63706000214828621402987371155, 1.77945837814196741939444883086, 1.79111576736196638703528689575, 1.84363909086548750036146894973, 1.98346944974395038378352768445, 1.99130131160294896058880834494, 2.14136880642560356831141629808, 2.37659138727350784648141047916, 2.41469179236214248949740474852, 2.42740358549198249010483388323, 2.44023882643598481388493392736, 2.52350246997799983206286316560

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

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