Properties

Label 20-189e10-1.1-c1e10-0-2
Degree $20$
Conductor $5.816\times 10^{22}$
Sign $1$
Analytic cond. $61.2899$
Root an. cond. $1.22848$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s + 3·7-s + 6·13-s + 5·16-s + 12·17-s + 3·19-s − 32·25-s − 9·28-s + 15·29-s − 9·31-s − 6·32-s + 6·37-s + 9·41-s + 3·43-s − 15·47-s − 7·49-s − 18·52-s + 9·53-s + 18·59-s + 12·61-s − 64-s − 10·67-s − 36·68-s + 3·73-s − 9·76-s + 20·79-s + 15·83-s + ⋯
L(s)  = 1  − 3/2·4-s + 1.13·7-s + 1.66·13-s + 5/4·16-s + 2.91·17-s + 0.688·19-s − 6.39·25-s − 1.70·28-s + 2.78·29-s − 1.61·31-s − 1.06·32-s + 0.986·37-s + 1.40·41-s + 0.457·43-s − 2.18·47-s − 49-s − 2.49·52-s + 1.23·53-s + 2.34·59-s + 1.53·61-s − 1/8·64-s − 1.22·67-s − 4.36·68-s + 0.351·73-s − 1.03·76-s + 2.25·79-s + 1.64·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(3^{30} \cdot 7^{10}\)
Sign: $1$
Analytic conductor: \(61.2899\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 3^{30} \cdot 7^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.859445761\)
\(L(\frac12)\) \(\approx\) \(1.859445761\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - 3 T + 16 T^{2} - 62 T^{3} + 220 T^{4} - 473 T^{5} + 220 p T^{6} - 62 p^{2} T^{7} + 16 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
good2 \( 1 + 3 T^{2} + p^{2} T^{4} + 3 p T^{5} - p T^{6} - 3 T^{7} - 7 p T^{8} - 39 T^{9} + T^{10} - 39 p T^{11} - 7 p^{3} T^{12} - 3 p^{3} T^{13} - p^{5} T^{14} + 3 p^{6} T^{15} + p^{8} T^{16} + 3 p^{8} T^{18} + p^{10} T^{20} \)
5 \( ( 1 + 16 T^{2} + 6 T^{3} + 127 T^{4} + 51 T^{5} + 127 p T^{6} + 6 p^{2} T^{7} + 16 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
11 \( 1 - 6 p T^{2} + 2257 T^{4} - 51461 T^{6} + 77732 p T^{8} - 10752323 T^{10} + 77732 p^{3} T^{12} - 51461 p^{4} T^{14} + 2257 p^{6} T^{16} - 6 p^{9} T^{18} + p^{10} T^{20} \)
13 \( 1 - 6 T + 68 T^{2} - 336 T^{3} + 2292 T^{4} - 723 p T^{5} + 51837 T^{6} - 187401 T^{7} + 909867 T^{8} - 3004662 T^{9} + 13054461 T^{10} - 3004662 p T^{11} + 909867 p^{2} T^{12} - 187401 p^{3} T^{13} + 51837 p^{4} T^{14} - 723 p^{6} T^{15} + 2292 p^{6} T^{16} - 336 p^{7} T^{17} + 68 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
17 \( 1 - 12 T + 26 T^{2} + 36 T^{3} + 1143 T^{4} - 5247 T^{5} - 21540 T^{6} + 73476 T^{7} + 337539 T^{8} + 599625 T^{9} - 13374333 T^{10} + 599625 p T^{11} + 337539 p^{2} T^{12} + 73476 p^{3} T^{13} - 21540 p^{4} T^{14} - 5247 p^{5} T^{15} + 1143 p^{6} T^{16} + 36 p^{7} T^{17} + 26 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 - 3 T + 71 T^{2} - 204 T^{3} + 2646 T^{4} - 8547 T^{5} + 74607 T^{6} - 258954 T^{7} + 1743726 T^{8} - 5989488 T^{9} + 35261703 T^{10} - 5989488 p T^{11} + 1743726 p^{2} T^{12} - 258954 p^{3} T^{13} + 74607 p^{4} T^{14} - 8547 p^{5} T^{15} + 2646 p^{6} T^{16} - 204 p^{7} T^{17} + 71 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
23 \( 1 - 165 T^{2} + 13144 T^{4} - 668429 T^{6} + 24050461 T^{8} - 639494549 T^{10} + 24050461 p^{2} T^{12} - 668429 p^{4} T^{14} + 13144 p^{6} T^{16} - 165 p^{8} T^{18} + p^{10} T^{20} \)
29 \( 1 - 15 T + 150 T^{2} - 1125 T^{3} + 6691 T^{4} - 30108 T^{5} + 81631 T^{6} + 232971 T^{7} - 5137202 T^{8} + 44535417 T^{9} - 275752187 T^{10} + 44535417 p T^{11} - 5137202 p^{2} T^{12} + 232971 p^{3} T^{13} + 81631 p^{4} T^{14} - 30108 p^{5} T^{15} + 6691 p^{6} T^{16} - 1125 p^{7} T^{17} + 150 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 + 9 T + 149 T^{2} + 1098 T^{3} + 10878 T^{4} + 60723 T^{5} + 461409 T^{6} + 2027286 T^{7} + 13421802 T^{8} + 50078664 T^{9} + 374531595 T^{10} + 50078664 p T^{11} + 13421802 p^{2} T^{12} + 2027286 p^{3} T^{13} + 461409 p^{4} T^{14} + 60723 p^{5} T^{15} + 10878 p^{6} T^{16} + 1098 p^{7} T^{17} + 149 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 - 6 T - 97 T^{2} + 194 T^{3} + 7179 T^{4} + 3556 T^{5} - 323794 T^{6} - 533292 T^{7} + 10739317 T^{8} + 10946526 T^{9} - 345629139 T^{10} + 10946526 p T^{11} + 10739317 p^{2} T^{12} - 533292 p^{3} T^{13} - 323794 p^{4} T^{14} + 3556 p^{5} T^{15} + 7179 p^{6} T^{16} + 194 p^{7} T^{17} - 97 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
41 \( 1 - 9 T - 34 T^{2} + 747 T^{3} - 2085 T^{4} - 20394 T^{5} + 110775 T^{6} + 15219 p T^{7} - 5992218 T^{8} - 18494757 T^{9} + 381591615 T^{10} - 18494757 p T^{11} - 5992218 p^{2} T^{12} + 15219 p^{4} T^{13} + 110775 p^{4} T^{14} - 20394 p^{5} T^{15} - 2085 p^{6} T^{16} + 747 p^{7} T^{17} - 34 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 - 3 T - 79 T^{2} + 1100 T^{3} + 1674 T^{4} - 79931 T^{5} + 324899 T^{6} + 75114 p T^{7} - 28512986 T^{8} - 52724394 T^{9} + 1438527201 T^{10} - 52724394 p T^{11} - 28512986 p^{2} T^{12} + 75114 p^{4} T^{13} + 324899 p^{4} T^{14} - 79931 p^{5} T^{15} + 1674 p^{6} T^{16} + 1100 p^{7} T^{17} - 79 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 + 15 T - 49 T^{2} - 1068 T^{3} + 9486 T^{4} + 83265 T^{5} - 760914 T^{6} - 61443 p T^{7} + 57371082 T^{8} + 59131839 T^{9} - 3026317959 T^{10} + 59131839 p T^{11} + 57371082 p^{2} T^{12} - 61443 p^{4} T^{13} - 760914 p^{4} T^{14} + 83265 p^{5} T^{15} + 9486 p^{6} T^{16} - 1068 p^{7} T^{17} - 49 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 - 9 T + 237 T^{2} - 1890 T^{3} + 28720 T^{4} - 208689 T^{5} + 2523571 T^{6} - 16852668 T^{7} + 178203742 T^{8} - 1088604978 T^{9} + 10361882797 T^{10} - 1088604978 p T^{11} + 178203742 p^{2} T^{12} - 16852668 p^{3} T^{13} + 2523571 p^{4} T^{14} - 208689 p^{5} T^{15} + 28720 p^{6} T^{16} - 1890 p^{7} T^{17} + 237 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 - 18 T - 55 T^{2} + 1536 T^{3} + 19971 T^{4} - 205494 T^{5} - 1764945 T^{6} + 8798931 T^{7} + 181121100 T^{8} - 308804295 T^{9} - 11121159681 T^{10} - 308804295 p T^{11} + 181121100 p^{2} T^{12} + 8798931 p^{3} T^{13} - 1764945 p^{4} T^{14} - 205494 p^{5} T^{15} + 19971 p^{6} T^{16} + 1536 p^{7} T^{17} - 55 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 - 12 T + 251 T^{2} - 2436 T^{3} + 34779 T^{4} - 347730 T^{5} + 3716049 T^{6} - 34819755 T^{7} + 301038894 T^{8} - 2709237273 T^{9} + 20488848807 T^{10} - 2709237273 p T^{11} + 301038894 p^{2} T^{12} - 34819755 p^{3} T^{13} + 3716049 p^{4} T^{14} - 347730 p^{5} T^{15} + 34779 p^{6} T^{16} - 2436 p^{7} T^{17} + 251 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 + 10 T - 182 T^{2} - 2448 T^{3} + 18867 T^{4} + 319605 T^{5} - 772530 T^{6} - 23213154 T^{7} - 12219093 T^{8} + 697872289 T^{9} + 3674653819 T^{10} + 697872289 p T^{11} - 12219093 p^{2} T^{12} - 23213154 p^{3} T^{13} - 772530 p^{4} T^{14} + 319605 p^{5} T^{15} + 18867 p^{6} T^{16} - 2448 p^{7} T^{17} - 182 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 - 351 T^{2} + 63037 T^{4} - 7800935 T^{6} + 747809113 T^{8} - 58386380555 T^{10} + 747809113 p^{2} T^{12} - 7800935 p^{4} T^{14} + 63037 p^{6} T^{16} - 351 p^{8} T^{18} + p^{10} T^{20} \)
73 \( 1 - 3 T + 260 T^{2} - 771 T^{3} + 36639 T^{4} - 155724 T^{5} + 3663555 T^{6} - 21043473 T^{7} + 293486934 T^{8} - 2074196103 T^{9} + 21799556757 T^{10} - 2074196103 p T^{11} + 293486934 p^{2} T^{12} - 21043473 p^{3} T^{13} + 3663555 p^{4} T^{14} - 155724 p^{5} T^{15} + 36639 p^{6} T^{16} - 771 p^{7} T^{17} + 260 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 - 20 T + 46 T^{2} - 144 T^{3} + 21153 T^{4} - 101181 T^{5} - 106944 T^{6} - 9264000 T^{7} + 7962453 T^{8} + 230795113 T^{9} + 4723714795 T^{10} + 230795113 p T^{11} + 7962453 p^{2} T^{12} - 9264000 p^{3} T^{13} - 106944 p^{4} T^{14} - 101181 p^{5} T^{15} + 21153 p^{6} T^{16} - 144 p^{7} T^{17} + 46 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 - 15 T - 136 T^{2} + 1773 T^{3} + 22674 T^{4} - 93717 T^{5} - 3687774 T^{6} + 10067337 T^{7} + 346135869 T^{8} - 496605294 T^{9} - 27460905396 T^{10} - 496605294 p T^{11} + 346135869 p^{2} T^{12} + 10067337 p^{3} T^{13} - 3687774 p^{4} T^{14} - 93717 p^{5} T^{15} + 22674 p^{6} T^{16} + 1773 p^{7} T^{17} - 136 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 + 24 T + 44 T^{2} - 2592 T^{3} - 1287 T^{4} + 278721 T^{5} - 235110 T^{6} - 13705920 T^{7} + 186157425 T^{8} + 904992183 T^{9} - 14602879521 T^{10} + 904992183 p T^{11} + 186157425 p^{2} T^{12} - 13705920 p^{3} T^{13} - 235110 p^{4} T^{14} + 278721 p^{5} T^{15} - 1287 p^{6} T^{16} - 2592 p^{7} T^{17} + 44 p^{8} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 - 6 T + 311 T^{2} - 1794 T^{3} + 51903 T^{4} - 385032 T^{5} + 6010353 T^{6} - 63309837 T^{7} + 574248354 T^{8} - 8264282925 T^{9} + 54719955099 T^{10} - 8264282925 p T^{11} + 574248354 p^{2} T^{12} - 63309837 p^{3} T^{13} + 6010353 p^{4} T^{14} - 385032 p^{5} T^{15} + 51903 p^{6} T^{16} - 1794 p^{7} T^{17} + 311 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \)
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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

−5.13613775594495937281790311806, −4.81912146376243006071843865895, −4.74891142152707397477515615259, −4.57382888118374017129140050805, −4.37370206069095027054862632734, −4.17653634802018374766490440875, −4.13049287557285688736249042805, −4.01699845085200827681155761476, −3.86184859608808813795456968732, −3.84047074365817117871382740445, −3.60483737371576070938052427901, −3.54475387748830543146082721901, −3.43304210968603055617101126965, −3.16196804717800642795217538994, −2.91865232577819247315551420581, −2.87875163801396666864982086374, −2.76903051701591595685329234915, −2.15634764867644485394192056760, −2.09670861751888765707811504253, −1.97917685175824363867946545380, −1.73982669211735289751978603091, −1.68298034256384422298002434118, −1.12281363211009344201097969591, −1.00265109611426784812380914614, −0.60589130754647243049971770521, 0.60589130754647243049971770521, 1.00265109611426784812380914614, 1.12281363211009344201097969591, 1.68298034256384422298002434118, 1.73982669211735289751978603091, 1.97917685175824363867946545380, 2.09670861751888765707811504253, 2.15634764867644485394192056760, 2.76903051701591595685329234915, 2.87875163801396666864982086374, 2.91865232577819247315551420581, 3.16196804717800642795217538994, 3.43304210968603055617101126965, 3.54475387748830543146082721901, 3.60483737371576070938052427901, 3.84047074365817117871382740445, 3.86184859608808813795456968732, 4.01699845085200827681155761476, 4.13049287557285688736249042805, 4.17653634802018374766490440875, 4.37370206069095027054862632734, 4.57382888118374017129140050805, 4.74891142152707397477515615259, 4.81912146376243006071843865895, 5.13613775594495937281790311806

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

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