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Properties

Label 20-370e10-1.1-c1e10-0-1
Degree 2020
Conductor 4.809×10254.809\times 10^{25}
Sign 11
Analytic cond. 50674.350674.3
Root an. cond. 1.718851.71885
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·2-s + 55·4-s − 3·5-s − 220·8-s + 11·9-s + 30·10-s − 2·13-s + 715·16-s + 18·17-s − 110·18-s − 165·20-s + 10·23-s + 7·25-s + 20·26-s − 2.00e3·32-s − 180·34-s + 605·36-s − 8·37-s + 660·40-s − 4·41-s − 10·43-s − 33·45-s − 100·46-s + 31·49-s − 70·50-s − 110·52-s + 5.00e3·64-s + ⋯
L(s)  = 1  − 7.07·2-s + 55/2·4-s − 1.34·5-s − 77.7·8-s + 11/3·9-s + 9.48·10-s − 0.554·13-s + 178.·16-s + 4.36·17-s − 25.9·18-s − 36.8·20-s + 2.08·23-s + 7/5·25-s + 3.92·26-s − 353.·32-s − 30.8·34-s + 100.·36-s − 1.31·37-s + 104.·40-s − 0.624·41-s − 1.52·43-s − 4.91·45-s − 14.7·46-s + 31/7·49-s − 9.89·50-s − 15.2·52-s + 625.·64-s + ⋯

Functional equation

Λ(s)=((210⋅510⋅3710)s/2 ΓC(s)10 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((210⋅510⋅3710)s/2 ΓC(s+1/2)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2020
Conductor: 210⋅510⋅37102^{10} \cdot 5^{10} \cdot 37^{10}
Sign: 11
Analytic conductor: 50674.350674.3
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (20, 210⋅510⋅3710, ( :[1/2]10), 1)(20,\ 2^{10} \cdot 5^{10} \cdot 37^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.19002484000.1900248400
L(12)L(\frac12) ≈\approx 0.19002484000.1900248400
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)
bad2 (1+T)10 ( 1 + T )^{10}
5 1+3T+2T2−16T3−19T4−22T5−19pT6−16p2T7+2p3T8+3p4T9+p5T10 1 + 3 T + 2 T^{2} - 16 T^{3} - 19 T^{4} - 22 T^{5} - 19 p T^{6} - 16 p^{2} T^{7} + 2 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10}
37 1+8T+97T2+576T3+5282T4+25584T5+5282pT6+576p2T7+97p3T8+8p4T9+p5T10 1 + 8 T + 97 T^{2} + 576 T^{3} + 5282 T^{4} + 25584 T^{5} + 5282 p T^{6} + 576 p^{2} T^{7} + 97 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10}
good3 1−11T2+52T4−32pT6−193T8+1390T10−193p2T12−32p5T14+52p6T16−11p8T18+p10T20 1 - 11 T^{2} + 52 T^{4} - 32 p T^{6} - 193 T^{8} + 1390 T^{10} - 193 p^{2} T^{12} - 32 p^{5} T^{14} + 52 p^{6} T^{16} - 11 p^{8} T^{18} + p^{10} T^{20}
7 1−31T2+459T4−4364T6+31924T8−218314T10+31924p2T12−4364p4T14+459p6T16−31p8T18+p10T20 1 - 31 T^{2} + 459 T^{4} - 4364 T^{6} + 31924 T^{8} - 218314 T^{10} + 31924 p^{2} T^{12} - 4364 p^{4} T^{14} + 459 p^{6} T^{16} - 31 p^{8} T^{18} + p^{10} T^{20}
11 (1+27T2+51T3+302T4+1074T5+302pT6+51p2T7+27p3T8+p5T10)2 ( 1 + 27 T^{2} + 51 T^{3} + 302 T^{4} + 1074 T^{5} + 302 p T^{6} + 51 p^{2} T^{7} + 27 p^{3} T^{8} + p^{5} T^{10} )^{2}
13 (1+T+2pT2−48T3+329T4−1098T5+329pT6−48p2T7+2p4T8+p4T9+p5T10)2 ( 1 + T + 2 p T^{2} - 48 T^{3} + 329 T^{4} - 1098 T^{5} + 329 p T^{6} - 48 p^{2} T^{7} + 2 p^{4} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2}
17 (1−9T+89T2−504T3+2982T4−12078T5+2982pT6−504p2T7+89p3T8−9p4T9+p5T10)2 ( 1 - 9 T + 89 T^{2} - 504 T^{3} + 2982 T^{4} - 12078 T^{5} + 2982 p T^{6} - 504 p^{2} T^{7} + 89 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2}
19 1−72T2+2977T4−90248T6+2204454T8−45625440T10+2204454p2T12−90248p4T14+2977p6T16−72p8T18+p10T20 1 - 72 T^{2} + 2977 T^{4} - 90248 T^{6} + 2204454 T^{8} - 45625440 T^{10} + 2204454 p^{2} T^{12} - 90248 p^{4} T^{14} + 2977 p^{6} T^{16} - 72 p^{8} T^{18} + p^{10} T^{20}
23 (1−5T+52T2−152T3+1199T4−2470T5+1199pT6−152p2T7+52p3T8−5p4T9+p5T10)2 ( 1 - 5 T + 52 T^{2} - 152 T^{3} + 1199 T^{4} - 2470 T^{5} + 1199 p T^{6} - 152 p^{2} T^{7} + 52 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} )^{2}
29 1−108T2+5879T4−231107T6+276700pT8−249440874T10+276700p3T12−231107p4T14+5879p6T16−108p8T18+p10T20 1 - 108 T^{2} + 5879 T^{4} - 231107 T^{6} + 276700 p T^{8} - 249440874 T^{10} + 276700 p^{3} T^{12} - 231107 p^{4} T^{14} + 5879 p^{6} T^{16} - 108 p^{8} T^{18} + p^{10} T^{20}
31 1−194T2+18979T4−1222911T6+57220962T8−2026943118T10+57220962p2T12−1222911p4T14+18979p6T16−194p8T18+p10T20 1 - 194 T^{2} + 18979 T^{4} - 1222911 T^{6} + 57220962 T^{8} - 2026943118 T^{10} + 57220962 p^{2} T^{12} - 1222911 p^{4} T^{14} + 18979 p^{6} T^{16} - 194 p^{8} T^{18} + p^{10} T^{20}
41 (1+2T+161T2+417T3+11390T4+27434T5+11390pT6+417p2T7+161p3T8+2p4T9+p5T10)2 ( 1 + 2 T + 161 T^{2} + 417 T^{3} + 11390 T^{4} + 27434 T^{5} + 11390 p T^{6} + 417 p^{2} T^{7} + 161 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2}
43 (1+5T+127T2+784T3+8994T4+46310T5+8994pT6+784p2T7+127p3T8+5p4T9+p5T10)2 ( 1 + 5 T + 127 T^{2} + 784 T^{3} + 8994 T^{4} + 46310 T^{5} + 8994 p T^{6} + 784 p^{2} T^{7} + 127 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} )^{2}
47 1−160T2+11993T4−511560T6+12029318T8−253577648T10+12029318p2T12−511560p4T14+11993p6T16−160p8T18+p10T20 1 - 160 T^{2} + 11993 T^{4} - 511560 T^{6} + 12029318 T^{8} - 253577648 T^{10} + 12029318 p^{2} T^{12} - 511560 p^{4} T^{14} + 11993 p^{6} T^{16} - 160 p^{8} T^{18} + p^{10} T^{20}
53 1−273T2+40401T4−4128228T6+317307718T8−18960642934T10+317307718p2T12−4128228p4T14+40401p6T16−273p8T18+p10T20 1 - 273 T^{2} + 40401 T^{4} - 4128228 T^{6} + 317307718 T^{8} - 18960642934 T^{10} + 317307718 p^{2} T^{12} - 4128228 p^{4} T^{14} + 40401 p^{6} T^{16} - 273 p^{8} T^{18} + p^{10} T^{20}
59 1−352T2+63153T4−7528424T6+659017222T8−44169143152T10+659017222p2T12−7528424p4T14+63153p6T16−352p8T18+p10T20 1 - 352 T^{2} + 63153 T^{4} - 7528424 T^{6} + 659017222 T^{8} - 44169143152 T^{10} + 659017222 p^{2} T^{12} - 7528424 p^{4} T^{14} + 63153 p^{6} T^{16} - 352 p^{8} T^{18} + p^{10} T^{20}
61 1−460T2+97647T4−12826483T6+19411236pT8−82282695178T10+19411236p3T12−12826483p4T14+97647p6T16−460p8T18+p10T20 1 - 460 T^{2} + 97647 T^{4} - 12826483 T^{6} + 19411236 p T^{8} - 82282695178 T^{10} + 19411236 p^{3} T^{12} - 12826483 p^{4} T^{14} + 97647 p^{6} T^{16} - 460 p^{8} T^{18} + p^{10} T^{20}
67 1−303T2+54020T4−6711456T6+641201935T8−48089295626T10+641201935p2T12−6711456p4T14+54020p6T16−303p8T18+p10T20 1 - 303 T^{2} + 54020 T^{4} - 6711456 T^{6} + 641201935 T^{8} - 48089295626 T^{10} + 641201935 p^{2} T^{12} - 6711456 p^{4} T^{14} + 54020 p^{6} T^{16} - 303 p^{8} T^{18} + p^{10} T^{20}
71 (1+10T+191T2+944T3+12662T4+37836T5+12662pT6+944p2T7+191p3T8+10p4T9+p5T10)2 ( 1 + 10 T + 191 T^{2} + 944 T^{3} + 12662 T^{4} + 37836 T^{5} + 12662 p T^{6} + 944 p^{2} T^{7} + 191 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} )^{2}
73 1−541T2+140394T4−23094490T6+2668314981T8−226082874802T10+2668314981p2T12−23094490p4T14+140394p6T16−541p8T18+p10T20 1 - 541 T^{2} + 140394 T^{4} - 23094490 T^{6} + 2668314981 T^{8} - 226082874802 T^{10} + 2668314981 p^{2} T^{12} - 23094490 p^{4} T^{14} + 140394 p^{6} T^{16} - 541 p^{8} T^{18} + p^{10} T^{20}
79 1−499T2+127952T4−21543420T6+2613057515T8−237224216810T10+2613057515p2T12−21543420p4T14+127952p6T16−499p8T18+p10T20 1 - 499 T^{2} + 127952 T^{4} - 21543420 T^{6} + 2613057515 T^{8} - 237224216810 T^{10} + 2613057515 p^{2} T^{12} - 21543420 p^{4} T^{14} + 127952 p^{6} T^{16} - 499 p^{8} T^{18} + p^{10} T^{20}
83 1−364T2+62793T4−7248824T6+680746366T8−58266464440T10+680746366p2T12−7248824p4T14+62793p6T16−364p8T18+p10T20 1 - 364 T^{2} + 62793 T^{4} - 7248824 T^{6} + 680746366 T^{8} - 58266464440 T^{10} + 680746366 p^{2} T^{12} - 7248824 p^{4} T^{14} + 62793 p^{6} T^{16} - 364 p^{8} T^{18} + p^{10} T^{20}
89 1−478T2+122989T4−21250184T6+2738250882T8−274126598068T10+2738250882p2T12−21250184p4T14+122989p6T16−478p8T18+p10T20 1 - 478 T^{2} + 122989 T^{4} - 21250184 T^{6} + 2738250882 T^{8} - 274126598068 T^{10} + 2738250882 p^{2} T^{12} - 21250184 p^{4} T^{14} + 122989 p^{6} T^{16} - 478 p^{8} T^{18} + p^{10} T^{20}
97 (1−T+307T2−636T3+45116T4−103398T5+45116pT6−636p2T7+307p3T8−p4T9+p5T10)2 ( 1 - T + 307 T^{2} - 636 T^{3} + 45116 T^{4} - 103398 T^{5} + 45116 p T^{6} - 636 p^{2} T^{7} + 307 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2}
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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

−4.17905450228621559789044926836, −4.05493135743222481555777632216, −4.04688652980808904983934830242, −3.93717686973128500184503330128, −3.66003666680008734498482460014, −3.64250923049939807967731791641, −3.34950857922547723215366095390, −3.25735311500383785121737984036, −3.12463151612735575852697414868, −3.09165077297633279601414564889, −2.87184479737424485641990223304, −2.80636339663185185620768448702, −2.68347517565279136380534429680, −2.39325109486815813608967620118, −2.18381892287677649437099199069, −1.90476899800056057084679979235, −1.80734505940661399190237106758, −1.79878270560593480113151527628, −1.61057951853498497082538792134, −1.20856738834502712407637389645, −1.15202431365447672837557696624, −1.09669338376445786986416008131, −0.986373115977527113288308057356, −0.66658622304901902266149138920, −0.39426706405684723631154924903, 0.39426706405684723631154924903, 0.66658622304901902266149138920, 0.986373115977527113288308057356, 1.09669338376445786986416008131, 1.15202431365447672837557696624, 1.20856738834502712407637389645, 1.61057951853498497082538792134, 1.79878270560593480113151527628, 1.80734505940661399190237106758, 1.90476899800056057084679979235, 2.18381892287677649437099199069, 2.39325109486815813608967620118, 2.68347517565279136380534429680, 2.80636339663185185620768448702, 2.87184479737424485641990223304, 3.09165077297633279601414564889, 3.12463151612735575852697414868, 3.25735311500383785121737984036, 3.34950857922547723215366095390, 3.64250923049939807967731791641, 3.66003666680008734498482460014, 3.93717686973128500184503330128, 4.04688652980808904983934830242, 4.05493135743222481555777632216, 4.17905450228621559789044926836

Graph of the ZZ-function along the critical line

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