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Properties

Label 20-585e10-1.1-c1e10-0-2
Degree 2020
Conductor 4.694×10274.694\times 10^{27}
Sign 11
Analytic cond. 4.94686×1064.94686\times 10^{6}
Root an. cond. 2.161302.16130
Motivic weight 11
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  + 2·2-s + 4·4-s − 10·5-s − 7-s + 4·8-s − 20·10-s + 8·11-s + 13-s − 2·14-s + 6·16-s + 4·19-s − 40·20-s + 16·22-s − 6·23-s + 55·25-s + 2·26-s − 4·28-s + 16·29-s + 18·31-s + 6·32-s + 10·35-s + 4·37-s + 8·38-s − 40·40-s + 6·41-s − 15·43-s + 32·44-s + ⋯
L(s)  = 1  + 1.41·2-s + 2·4-s − 4.47·5-s − 0.377·7-s + 1.41·8-s − 6.32·10-s + 2.41·11-s + 0.277·13-s − 0.534·14-s + 3/2·16-s + 0.917·19-s − 8.94·20-s + 3.41·22-s − 1.25·23-s + 11·25-s + 0.392·26-s − 0.755·28-s + 2.97·29-s + 3.23·31-s + 1.06·32-s + 1.69·35-s + 0.657·37-s + 1.29·38-s − 6.32·40-s + 0.937·41-s − 2.28·43-s + 4.82·44-s + ⋯

Functional equation

Λ(s)=((320⋅510⋅1310)s/2 ΓC(s)10 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((320⋅510⋅1310)s/2 ΓC(s+1/2)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2020
Conductor: 320⋅510⋅13103^{20} \cdot 5^{10} \cdot 13^{10}
Sign: 11
Analytic conductor: 4.94686×1064.94686\times 10^{6}
Root analytic conductor: 2.161302.16130
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (20, 320⋅510⋅1310, ( :[1/2]10), 1)(20,\ 3^{20} \cdot 5^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 11.6192111411.61921114
L(12)L(\frac12) ≈\approx 11.6192111411.61921114
L(32)L(\frac{3}{2}) 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 1
5 (1+T)10 ( 1 + T )^{10}
13 1−T−21T2+106T3+185T4−2013T5+185pT6+106p2T7−21p3T8−p4T9+p5T10 1 - T - 21 T^{2} + 106 T^{3} + 185 T^{4} - 2013 T^{5} + 185 p T^{6} + 106 p^{2} T^{7} - 21 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10}
good2 1−pT+p2T3−3pT4+pT5+p3T6−3p2T7+p2T9+3p2T10+p3T11−3p5T13+p7T14+p6T15−3p7T16+p9T17−p10T19+p10T20 1 - p T + p^{2} T^{3} - 3 p T^{4} + p T^{5} + p^{3} T^{6} - 3 p^{2} T^{7} + p^{2} T^{9} + 3 p^{2} T^{10} + p^{3} T^{11} - 3 p^{5} T^{13} + p^{7} T^{14} + p^{6} T^{15} - 3 p^{7} T^{16} + p^{9} T^{17} - p^{10} T^{19} + p^{10} T^{20}
7 1+T−12T2+T3+88T4−37T5+76T6+359T7−5281T8−4p2T9+8116pT10−4p3T11−5281p2T12+359p3T13+76p4T14−37p5T15+88p6T16+p7T17−12p8T18+p9T19+p10T20 1 + T - 12 T^{2} + T^{3} + 88 T^{4} - 37 T^{5} + 76 T^{6} + 359 T^{7} - 5281 T^{8} - 4 p^{2} T^{9} + 8116 p T^{10} - 4 p^{3} T^{11} - 5281 p^{2} T^{12} + 359 p^{3} T^{13} + 76 p^{4} T^{14} - 37 p^{5} T^{15} + 88 p^{6} T^{16} + p^{7} T^{17} - 12 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20}
11 1−8T+9T2+136T3−537T4−136T5+6038T6−21216T7+31725T8+137272T9−918177T10+137272pT11+31725p2T12−21216p3T13+6038p4T14−136p5T15−537p6T16+136p7T17+9p8T18−8p9T19+p10T20 1 - 8 T + 9 T^{2} + 136 T^{3} - 537 T^{4} - 136 T^{5} + 6038 T^{6} - 21216 T^{7} + 31725 T^{8} + 137272 T^{9} - 918177 T^{10} + 137272 p T^{11} + 31725 p^{2} T^{12} - 21216 p^{3} T^{13} + 6038 p^{4} T^{14} - 136 p^{5} T^{15} - 537 p^{6} T^{16} + 136 p^{7} T^{17} + 9 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20}
17 1−35T2+28T3+315T4−324T5−42pT6−28406T7+117019T8+442610T9−3489795T10+442610pT11+117019p2T12−28406p3T13−42p5T14−324p5T15+315p6T16+28p7T17−35p8T18+p10T20 1 - 35 T^{2} + 28 T^{3} + 315 T^{4} - 324 T^{5} - 42 p T^{6} - 28406 T^{7} + 117019 T^{8} + 442610 T^{9} - 3489795 T^{10} + 442610 p T^{11} + 117019 p^{2} T^{12} - 28406 p^{3} T^{13} - 42 p^{5} T^{14} - 324 p^{5} T^{15} + 315 p^{6} T^{16} + 28 p^{7} T^{17} - 35 p^{8} T^{18} + p^{10} T^{20}
19 1−4T−71T2+188T3+3347T4−5424T5−5882pT6+91244T7+2934257T8−702040T9−61778793T10−702040pT11+2934257p2T12+91244p3T13−5882p5T14−5424p5T15+3347p6T16+188p7T17−71p8T18−4p9T19+p10T20 1 - 4 T - 71 T^{2} + 188 T^{3} + 3347 T^{4} - 5424 T^{5} - 5882 p T^{6} + 91244 T^{7} + 2934257 T^{8} - 702040 T^{9} - 61778793 T^{10} - 702040 p T^{11} + 2934257 p^{2} T^{12} + 91244 p^{3} T^{13} - 5882 p^{5} T^{14} - 5424 p^{5} T^{15} + 3347 p^{6} T^{16} + 188 p^{7} T^{17} - 71 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20}
23 1+6T−59T2−462T3+1899T4+17976T5−35110T6−381456T7+619997T8+3636078T9−11442537T10+3636078pT11+619997p2T12−381456p3T13−35110p4T14+17976p5T15+1899p6T16−462p7T17−59p8T18+6p9T19+p10T20 1 + 6 T - 59 T^{2} - 462 T^{3} + 1899 T^{4} + 17976 T^{5} - 35110 T^{6} - 381456 T^{7} + 619997 T^{8} + 3636078 T^{9} - 11442537 T^{10} + 3636078 p T^{11} + 619997 p^{2} T^{12} - 381456 p^{3} T^{13} - 35110 p^{4} T^{14} + 17976 p^{5} T^{15} + 1899 p^{6} T^{16} - 462 p^{7} T^{17} - 59 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20}
29 1−16T+53T2+280T3−3pT4−14930T5+13984T6−100288T7+3893633T8−2571612T9−117536991T10−2571612pT11+3893633p2T12−100288p3T13+13984p4T14−14930p5T15−3p7T16+280p7T17+53p8T18−16p9T19+p10T20 1 - 16 T + 53 T^{2} + 280 T^{3} - 3 p T^{4} - 14930 T^{5} + 13984 T^{6} - 100288 T^{7} + 3893633 T^{8} - 2571612 T^{9} - 117536991 T^{10} - 2571612 p T^{11} + 3893633 p^{2} T^{12} - 100288 p^{3} T^{13} + 13984 p^{4} T^{14} - 14930 p^{5} T^{15} - 3 p^{7} T^{16} + 280 p^{7} T^{17} + 53 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20}
31 (1−9T+125T2−938T3+7513T4−40255T5+7513pT6−938p2T7+125p3T8−9p4T9+p5T10)2 ( 1 - 9 T + 125 T^{2} - 938 T^{3} + 7513 T^{4} - 40255 T^{5} + 7513 p T^{6} - 938 p^{2} T^{7} + 125 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2}
37 1−4T−105T2+812T3+5223T4−60656T5−48342T6+2598544T7−6880531T8−41122020T9+443418561T10−41122020pT11−6880531p2T12+2598544p3T13−48342p4T14−60656p5T15+5223p6T16+812p7T17−105p8T18−4p9T19+p10T20 1 - 4 T - 105 T^{2} + 812 T^{3} + 5223 T^{4} - 60656 T^{5} - 48342 T^{6} + 2598544 T^{7} - 6880531 T^{8} - 41122020 T^{9} + 443418561 T^{10} - 41122020 p T^{11} - 6880531 p^{2} T^{12} + 2598544 p^{3} T^{13} - 48342 p^{4} T^{14} - 60656 p^{5} T^{15} + 5223 p^{6} T^{16} + 812 p^{7} T^{17} - 105 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20}
41 1−6T−71T2+198T3+2037T4+11598T5−152896T6+303432T7+7372433T8−28857882T9−198785607T10−28857882pT11+7372433p2T12+303432p3T13−152896p4T14+11598p5T15+2037p6T16+198p7T17−71p8T18−6p9T19+p10T20 1 - 6 T - 71 T^{2} + 198 T^{3} + 2037 T^{4} + 11598 T^{5} - 152896 T^{6} + 303432 T^{7} + 7372433 T^{8} - 28857882 T^{9} - 198785607 T^{10} - 28857882 p T^{11} + 7372433 p^{2} T^{12} + 303432 p^{3} T^{13} - 152896 p^{4} T^{14} + 11598 p^{5} T^{15} + 2037 p^{6} T^{16} + 198 p^{7} T^{17} - 71 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20}
43 1+15T−20T2−569T3+8062T4+40863T5−572708T6−1600281T7+26293173T8−7330552T9−1511025092T10−7330552pT11+26293173p2T12−1600281p3T13−572708p4T14+40863p5T15+8062p6T16−569p7T17−20p8T18+15p9T19+p10T20 1 + 15 T - 20 T^{2} - 569 T^{3} + 8062 T^{4} + 40863 T^{5} - 572708 T^{6} - 1600281 T^{7} + 26293173 T^{8} - 7330552 T^{9} - 1511025092 T^{10} - 7330552 p T^{11} + 26293173 p^{2} T^{12} - 1600281 p^{3} T^{13} - 572708 p^{4} T^{14} + 40863 p^{5} T^{15} + 8062 p^{6} T^{16} - 569 p^{7} T^{17} - 20 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20}
47 (1+10T+145T2+24pT3+10018T4+69286T5+10018pT6+24p3T7+145p3T8+10p4T9+p5T10)2 ( 1 + 10 T + 145 T^{2} + 24 p T^{3} + 10018 T^{4} + 69286 T^{5} + 10018 p T^{6} + 24 p^{3} T^{7} + 145 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} )^{2}
53 (1−20T+5pT2−2368T3+17722T4−123096T5+17722pT6−2368p2T7+5p4T8−20p4T9+p5T10)2 ( 1 - 20 T + 5 p T^{2} - 2368 T^{3} + 17722 T^{4} - 123096 T^{5} + 17722 p T^{6} - 2368 p^{2} T^{7} + 5 p^{4} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} )^{2}
59 1−12T−109T2+1836T3+7077T4−158166T5−285300T6+9000432T7+1945509T8−240199128T9+613215327T10−240199128pT11+1945509p2T12+9000432p3T13−285300p4T14−158166p5T15+7077p6T16+1836p7T17−109p8T18−12p9T19+p10T20 1 - 12 T - 109 T^{2} + 1836 T^{3} + 7077 T^{4} - 158166 T^{5} - 285300 T^{6} + 9000432 T^{7} + 1945509 T^{8} - 240199128 T^{9} + 613215327 T^{10} - 240199128 p T^{11} + 1945509 p^{2} T^{12} + 9000432 p^{3} T^{13} - 285300 p^{4} T^{14} - 158166 p^{5} T^{15} + 7077 p^{6} T^{16} + 1836 p^{7} T^{17} - 109 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20}
61 1+11T−170T2−1355T3+25322T4+107353T5−2797432T6−6232727T7+234614309T8+184344598T9−15525735436T10+184344598pT11+234614309p2T12−6232727p3T13−2797432p4T14+107353p5T15+25322p6T16−1355p7T17−170p8T18+11p9T19+p10T20 1 + 11 T - 170 T^{2} - 1355 T^{3} + 25322 T^{4} + 107353 T^{5} - 2797432 T^{6} - 6232727 T^{7} + 234614309 T^{8} + 184344598 T^{9} - 15525735436 T^{10} + 184344598 p T^{11} + 234614309 p^{2} T^{12} - 6232727 p^{3} T^{13} - 2797432 p^{4} T^{14} + 107353 p^{5} T^{15} + 25322 p^{6} T^{16} - 1355 p^{7} T^{17} - 170 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20}
67 1+5T−266T2−1217T3+41876T4+161677T5−4557484T6−11713703T7+393911771T8+360277294T9−28368034144T10+360277294pT11+393911771p2T12−11713703p3T13−4557484p4T14+161677p5T15+41876p6T16−1217p7T17−266p8T18+5p9T19+p10T20 1 + 5 T - 266 T^{2} - 1217 T^{3} + 41876 T^{4} + 161677 T^{5} - 4557484 T^{6} - 11713703 T^{7} + 393911771 T^{8} + 360277294 T^{9} - 28368034144 T^{10} + 360277294 p T^{11} + 393911771 p^{2} T^{12} - 11713703 p^{3} T^{13} - 4557484 p^{4} T^{14} + 161677 p^{5} T^{15} + 41876 p^{6} T^{16} - 1217 p^{7} T^{17} - 266 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20}
71 1−10T−233T2+1642T3+40821T4−175708T5−4974832T6+10835286T7+479818097T8−291547112T9−37711050573T10−291547112pT11+479818097p2T12+10835286p3T13−4974832p4T14−175708p5T15+40821p6T16+1642p7T17−233p8T18−10p9T19+p10T20 1 - 10 T - 233 T^{2} + 1642 T^{3} + 40821 T^{4} - 175708 T^{5} - 4974832 T^{6} + 10835286 T^{7} + 479818097 T^{8} - 291547112 T^{9} - 37711050573 T^{10} - 291547112 p T^{11} + 479818097 p^{2} T^{12} + 10835286 p^{3} T^{13} - 4974832 p^{4} T^{14} - 175708 p^{5} T^{15} + 40821 p^{6} T^{16} + 1642 p^{7} T^{17} - 233 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20}
73 (1−T+129T2−148T3+13711T4−1257T5+13711pT6−148p2T7+129p3T8−p4T9+p5T10)2 ( 1 - T + 129 T^{2} - 148 T^{3} + 13711 T^{4} - 1257 T^{5} + 13711 p T^{6} - 148 p^{2} T^{7} + 129 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2}
79 (1+17T+369T2+4286T3+54349T4+467343T5+54349pT6+4286p2T7+369p3T8+17p4T9+p5T10)2 ( 1 + 17 T + 369 T^{2} + 4286 T^{3} + 54349 T^{4} + 467343 T^{5} + 54349 p T^{6} + 4286 p^{2} T^{7} + 369 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} )^{2}
83 (1+16T+299T2+3252T3+38638T4+325648T5+38638pT6+3252p2T7+299p3T8+16p4T9+p5T10)2 ( 1 + 16 T + 299 T^{2} + 3252 T^{3} + 38638 T^{4} + 325648 T^{5} + 38638 p T^{6} + 3252 p^{2} T^{7} + 299 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} )^{2}
89 1−4T−327T2+1320T3+60217T4−221568T5−7481300T6+21676602T7+726349165T8−826863906T9−65088711959T10−826863906pT11+726349165p2T12+21676602p3T13−7481300p4T14−221568p5T15+60217p6T16+1320p7T17−327p8T18−4p9T19+p10T20 1 - 4 T - 327 T^{2} + 1320 T^{3} + 60217 T^{4} - 221568 T^{5} - 7481300 T^{6} + 21676602 T^{7} + 726349165 T^{8} - 826863906 T^{9} - 65088711959 T^{10} - 826863906 p T^{11} + 726349165 p^{2} T^{12} + 21676602 p^{3} T^{13} - 7481300 p^{4} T^{14} - 221568 p^{5} T^{15} + 60217 p^{6} T^{16} + 1320 p^{7} T^{17} - 327 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20}
97 1−11T−234T2+419T3+54134T4+86137T5−5323566T6−30313009T7+338701565T8+1490050320T9−15821303860T10+1490050320pT11+338701565p2T12−30313009p3T13−5323566p4T14+86137p5T15+54134p6T16+419p7T17−234p8T18−11p9T19+p10T20 1 - 11 T - 234 T^{2} + 419 T^{3} + 54134 T^{4} + 86137 T^{5} - 5323566 T^{6} - 30313009 T^{7} + 338701565 T^{8} + 1490050320 T^{9} - 15821303860 T^{10} + 1490050320 p T^{11} + 338701565 p^{2} T^{12} - 30313009 p^{3} T^{13} - 5323566 p^{4} T^{14} + 86137 p^{5} T^{15} + 54134 p^{6} T^{16} + 419 p^{7} T^{17} - 234 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20}
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   L(s)=∏p ∏j=120(1−αj,p p−s)−1L(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

−3.92077195501973632658157754023, −3.82114626857104250181853071340, −3.75801662527586584375552614411, −3.59898451390508016439093050041, −3.54553934900317029430149395806, −3.45962494245081517311096089939, −3.39612450331145986062101219649, −3.23577558499124725871142205990, −3.12398047394544547838782229573, −3.02432461290945445436867528633, −2.95519764124746822175091904104, −2.76070713127332083937670663946, −2.64131739105933418185145271968, −2.55963128393530351015437525547, −2.22218455976849604029564765924, −2.21375426964418622656186514437, −2.02439802509190014260273124543, −1.68522764645946517728936713467, −1.68271710606772950029574643307, −1.25160711772903179486485181586, −1.10492623093241095047129026657, −0.929091680662446651858634592799, −0.75487914430656409040485524777, −0.68718210871169463147620788087, −0.41813823383855153823387318029, 0.41813823383855153823387318029, 0.68718210871169463147620788087, 0.75487914430656409040485524777, 0.929091680662446651858634592799, 1.10492623093241095047129026657, 1.25160711772903179486485181586, 1.68271710606772950029574643307, 1.68522764645946517728936713467, 2.02439802509190014260273124543, 2.21375426964418622656186514437, 2.22218455976849604029564765924, 2.55963128393530351015437525547, 2.64131739105933418185145271968, 2.76070713127332083937670663946, 2.95519764124746822175091904104, 3.02432461290945445436867528633, 3.12398047394544547838782229573, 3.23577558499124725871142205990, 3.39612450331145986062101219649, 3.45962494245081517311096089939, 3.54553934900317029430149395806, 3.59898451390508016439093050041, 3.75801662527586584375552614411, 3.82114626857104250181853071340, 3.92077195501973632658157754023

Graph of the ZZ-function along the critical line

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