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Properties

Label 20-810e10-1.1-c3e10-0-1
Degree 2020
Conductor 1.216×10291.216\times 10^{29}
Sign 11
Analytic cond. 6.21599×10166.21599\times 10^{16}
Root an. cond. 6.913146.91314
Motivic weight 33
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  − 20·4-s + 22·5-s − 28·11-s + 240·16-s − 42·19-s − 440·20-s + 290·25-s + 216·29-s + 88·31-s + 38·41-s + 560·44-s + 989·49-s − 616·55-s − 2.05e3·59-s + 1.06e3·61-s − 2.24e3·64-s − 2.84e3·71-s + 840·76-s − 1.60e3·79-s + 5.28e3·80-s + 1.13e3·89-s − 924·95-s − 5.80e3·100-s + 4.19e3·101-s − 1.74e3·109-s − 4.32e3·116-s − 3.77e3·121-s + ⋯
L(s)  = 1  − 5/2·4-s + 1.96·5-s − 0.767·11-s + 15/4·16-s − 0.507·19-s − 4.91·20-s + 2.31·25-s + 1.38·29-s + 0.509·31-s + 0.144·41-s + 1.91·44-s + 2.88·49-s − 1.51·55-s − 4.52·59-s + 2.23·61-s − 4.37·64-s − 4.75·71-s + 1.26·76-s − 2.29·79-s + 7.37·80-s + 1.34·89-s − 0.997·95-s − 5.79·100-s + 4.12·101-s − 1.53·109-s − 3.45·116-s − 2.83·121-s + ⋯

Functional equation

Λ(s)=((210⋅340⋅510)s/2 ΓC(s)10 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{40} \cdot 5^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((210⋅340⋅510)s/2 ΓC(s+3/2)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{40} \cdot 5^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2020
Conductor: 210⋅340⋅5102^{10} \cdot 3^{40} \cdot 5^{10}
Sign: 11
Analytic conductor: 6.21599×10166.21599\times 10^{16}
Root analytic conductor: 6.913146.91314
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (20, 210⋅340⋅510, ( :[3/2]10), 1)(20,\ 2^{10} \cdot 3^{40} \cdot 5^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 37.9611291337.96112913
L(12)L(\frac12) ≈\approx 37.9611291337.96112913
L(52)L(\frac{5}{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+p2T2)5 ( 1 + p^{2} T^{2} )^{5}
3 1 1
5 1−22T+194T2−1968T3+4801pT4−9956p2T5+4801p4T6−1968p6T7+194p9T8−22p12T9+p15T10 1 - 22 T + 194 T^{2} - 1968 T^{3} + 4801 p T^{4} - 9956 p^{2} T^{5} + 4801 p^{4} T^{6} - 1968 p^{6} T^{7} + 194 p^{9} T^{8} - 22 p^{12} T^{9} + p^{15} T^{10}
good7 1−989T2+11253p2T4−216725100T6+82684409322T8−597847827342p2T10+82684409322p6T12−216725100p12T14+11253p20T16−989p24T18+p30T20 1 - 989 T^{2} + 11253 p^{2} T^{4} - 216725100 T^{6} + 82684409322 T^{8} - 597847827342 p^{2} T^{10} + 82684409322 p^{6} T^{12} - 216725100 p^{12} T^{14} + 11253 p^{20} T^{16} - 989 p^{24} T^{18} + p^{30} T^{20}
11 (1+14T+2180T2+45636T3+4259435T4+28129324T5+4259435p3T6+45636p6T7+2180p9T8+14p12T9+p15T10)2 ( 1 + 14 T + 2180 T^{2} + 45636 T^{3} + 4259435 T^{4} + 28129324 T^{5} + 4259435 p^{3} T^{6} + 45636 p^{6} T^{7} + 2180 p^{9} T^{8} + 14 p^{12} T^{9} + p^{15} T^{10} )^{2}
13 1−8467T2+41346012T4−11961676089pT6+462940014645519T8−1116082425021637272T10+462940014645519p6T12−11961676089p13T14+41346012p18T16−8467p24T18+p30T20 1 - 8467 T^{2} + 41346012 T^{4} - 11961676089 p T^{6} + 462940014645519 T^{8} - 1116082425021637272 T^{10} + 462940014645519 p^{6} T^{12} - 11961676089 p^{13} T^{14} + 41346012 p^{18} T^{16} - 8467 p^{24} T^{18} + p^{30} T^{20}
17 1−27564T2+387966374T4−215798168622pT6+25966368140009113T8− 1 - 27564 T^{2} + 387966374 T^{4} - 215798168622 p T^{6} + 25966368140009113 T^{8} - 14 ⁣⋯ ⁣0414\!\cdots\!04T10+25966368140009113p6T12−215798168622p13T14+387966374p18T16−27564p24T18+p30T20 T^{10} + 25966368140009113 p^{6} T^{12} - 215798168622 p^{13} T^{14} + 387966374 p^{18} T^{16} - 27564 p^{24} T^{18} + p^{30} T^{20}
19 (1+21T+14570T2+323139T3+157538305T4+2409635700T5+157538305p3T6+323139p6T7+14570p9T8+21p12T9+p15T10)2 ( 1 + 21 T + 14570 T^{2} + 323139 T^{3} + 157538305 T^{4} + 2409635700 T^{5} + 157538305 p^{3} T^{6} + 323139 p^{6} T^{7} + 14570 p^{9} T^{8} + 21 p^{12} T^{9} + p^{15} T^{10} )^{2}
23 1−73005T2+2524256189T4−56433033194028T6+941157412042844530T8− 1 - 73005 T^{2} + 2524256189 T^{4} - 56433033194028 T^{6} + 941157412042844530 T^{8} - 12 ⁣⋯ ⁣5412\!\cdots\!54T10+941157412042844530p6T12−56433033194028p12T14+2524256189p18T16−73005p24T18+p30T20 T^{10} + 941157412042844530 p^{6} T^{12} - 56433033194028 p^{12} T^{14} + 2524256189 p^{18} T^{16} - 73005 p^{24} T^{18} + p^{30} T^{20}
29 (1−108T+93091T2−7346268T3+3874520917T4−237257418168T5+3874520917p3T6−7346268p6T7+93091p9T8−108p12T9+p15T10)2 ( 1 - 108 T + 93091 T^{2} - 7346268 T^{3} + 3874520917 T^{4} - 237257418168 T^{5} + 3874520917 p^{3} T^{6} - 7346268 p^{6} T^{7} + 93091 p^{9} T^{8} - 108 p^{12} T^{9} + p^{15} T^{10} )^{2}
31 (1−44T+87240T2+2084718T3+3208376163T4+199557551580T5+3208376163p3T6+2084718p6T7+87240p9T8−44p12T9+p15T10)2 ( 1 - 44 T + 87240 T^{2} + 2084718 T^{3} + 3208376163 T^{4} + 199557551580 T^{5} + 3208376163 p^{3} T^{6} + 2084718 p^{6} T^{7} + 87240 p^{9} T^{8} - 44 p^{12} T^{9} + p^{15} T^{10} )^{2}
37 1−436280T2+88749670398T4−11064509169435714T6+ 1 - 436280 T^{2} + 88749670398 T^{4} - 11064509169435714 T^{6} + 93 ⁣⋯ ⁣2193\!\cdots\!21T8− T^{8} - 55 ⁣⋯ ⁣9255\!\cdots\!92T10+ T^{10} + 93 ⁣⋯ ⁣2193\!\cdots\!21p6T12−11064509169435714p12T14+88749670398p18T16−436280p24T18+p30T20 p^{6} T^{12} - 11064509169435714 p^{12} T^{14} + 88749670398 p^{18} T^{16} - 436280 p^{24} T^{18} + p^{30} T^{20}
41 (1−19T+104036T2+27771483T3+5625867263T4+2838087609352T5+5625867263p3T6+27771483p6T7+104036p9T8−19p12T9+p15T10)2 ( 1 - 19 T + 104036 T^{2} + 27771483 T^{3} + 5625867263 T^{4} + 2838087609352 T^{5} + 5625867263 p^{3} T^{6} + 27771483 p^{6} T^{7} + 104036 p^{9} T^{8} - 19 p^{12} T^{9} + p^{15} T^{10} )^{2}
43 1−547898T2+142594310133T4−23619463504397688T6+ 1 - 547898 T^{2} + 142594310133 T^{4} - 23619463504397688 T^{6} + 28 ⁣⋯ ⁣5428\!\cdots\!54T8− T^{8} - 25 ⁣⋯ ⁣8825\!\cdots\!88T10+ T^{10} + 28 ⁣⋯ ⁣5428\!\cdots\!54p6T12−23619463504397688p12T14+142594310133p18T16−547898p24T18+p30T20 p^{6} T^{12} - 23619463504397688 p^{12} T^{14} + 142594310133 p^{18} T^{16} - 547898 p^{24} T^{18} + p^{30} T^{20}
47 1−160917T2+32408020925T4−3943665433054572T6+ 1 - 160917 T^{2} + 32408020925 T^{4} - 3943665433054572 T^{6} + 59 ⁣⋯ ⁣7059\!\cdots\!70T8− T^{8} - 57 ⁣⋯ ⁣8257\!\cdots\!82T10+ T^{10} + 59 ⁣⋯ ⁣7059\!\cdots\!70p6T12−3943665433054572p12T14+32408020925p18T16−160917p24T18+p30T20 p^{6} T^{12} - 3943665433054572 p^{12} T^{14} + 32408020925 p^{18} T^{16} - 160917 p^{24} T^{18} + p^{30} T^{20}
53 1−481721T2+124883352789T4−26025554285726556T6+ 1 - 481721 T^{2} + 124883352789 T^{4} - 26025554285726556 T^{6} + 48 ⁣⋯ ⁣3848\!\cdots\!38T8− T^{8} - 78 ⁣⋯ ⁣2678\!\cdots\!26T10+ T^{10} + 48 ⁣⋯ ⁣3848\!\cdots\!38p6T12−26025554285726556p12T14+124883352789p18T16−481721p24T18+p30T20 p^{6} T^{12} - 26025554285726556 p^{12} T^{14} + 124883352789 p^{18} T^{16} - 481721 p^{24} T^{18} + p^{30} T^{20}
59 (1+1025T+934574T2+580021995T3+352700074433T4+162166394208160T5+352700074433p3T6+580021995p6T7+934574p9T8+1025p12T9+p15T10)2 ( 1 + 1025 T + 934574 T^{2} + 580021995 T^{3} + 352700074433 T^{4} + 162166394208160 T^{5} + 352700074433 p^{3} T^{6} + 580021995 p^{6} T^{7} + 934574 p^{9} T^{8} + 1025 p^{12} T^{9} + p^{15} T^{10} )^{2}
61 (1−532T+39126T2+81398814T3+68239959321T4−58390875038148T5+68239959321p3T6+81398814p6T7+39126p9T8−532p12T9+p15T10)2 ( 1 - 532 T + 39126 T^{2} + 81398814 T^{3} + 68239959321 T^{4} - 58390875038148 T^{5} + 68239959321 p^{3} T^{6} + 81398814 p^{6} T^{7} + 39126 p^{9} T^{8} - 532 p^{12} T^{9} + p^{15} T^{10} )^{2}
67 1−1586630T2+1379172381045T4−812854783643216040T6+ 1 - 1586630 T^{2} + 1379172381045 T^{4} - 812854783643216040 T^{6} + 35 ⁣⋯ ⁣3035\!\cdots\!30T8− T^{8} - 12 ⁣⋯ ⁣1212\!\cdots\!12T10+ T^{10} + 35 ⁣⋯ ⁣3035\!\cdots\!30p6T12−812854783643216040p12T14+1379172381045p18T16−1586630p24T18+p30T20 p^{6} T^{12} - 812854783643216040 p^{12} T^{14} + 1379172381045 p^{18} T^{16} - 1586630 p^{24} T^{18} + p^{30} T^{20}
71 (1+1422T+1367800T2+667635588T3+237071942839T4+39943322814060T5+237071942839p3T6+667635588p6T7+1367800p9T8+1422p12T9+p15T10)2 ( 1 + 1422 T + 1367800 T^{2} + 667635588 T^{3} + 237071942839 T^{4} + 39943322814060 T^{5} + 237071942839 p^{3} T^{6} + 667635588 p^{6} T^{7} + 1367800 p^{9} T^{8} + 1422 p^{12} T^{9} + p^{15} T^{10} )^{2}
73 1−1686776T2+1611760751694T4−1069814961419818746T6+ 1 - 1686776 T^{2} + 1611760751694 T^{4} - 1069814961419818746 T^{6} + 55 ⁣⋯ ⁣9355\!\cdots\!93T8− T^{8} - 23 ⁣⋯ ⁣9623\!\cdots\!96T10+ T^{10} + 55 ⁣⋯ ⁣9355\!\cdots\!93p6T12−1069814961419818746p12T14+1611760751694p18T16−1686776p24T18+p30T20 p^{6} T^{12} - 1069814961419818746 p^{12} T^{14} + 1611760751694 p^{18} T^{16} - 1686776 p^{24} T^{18} + p^{30} T^{20}
79 (1+804T+1138415T2+981795360T3+809210827198T4+716956068535416T5+809210827198p3T6+981795360p6T7+1138415p9T8+804p12T9+p15T10)2 ( 1 + 804 T + 1138415 T^{2} + 981795360 T^{3} + 809210827198 T^{4} + 716956068535416 T^{5} + 809210827198 p^{3} T^{6} + 981795360 p^{6} T^{7} + 1138415 p^{9} T^{8} + 804 p^{12} T^{9} + p^{15} T^{10} )^{2}
83 1−4865814T2+11000705361557T4−15256377151494541416T6+ 1 - 4865814 T^{2} + 11000705361557 T^{4} - 15256377151494541416 T^{6} + 14 ⁣⋯ ⁣1414\!\cdots\!14T8− T^{8} - 96 ⁣⋯ ⁣0096\!\cdots\!00T10+ T^{10} + 14 ⁣⋯ ⁣1414\!\cdots\!14p6T12−15256377151494541416p12T14+11000705361557p18T16−4865814p24T18+p30T20 p^{6} T^{12} - 15256377151494541416 p^{12} T^{14} + 11000705361557 p^{18} T^{16} - 4865814 p^{24} T^{18} + p^{30} T^{20}
89 (1−566T+1553291T2+305869032T3+513050516045T4+915246283006106T5+513050516045p3T6+305869032p6T7+1553291p9T8−566p12T9+p15T10)2 ( 1 - 566 T + 1553291 T^{2} + 305869032 T^{3} + 513050516045 T^{4} + 915246283006106 T^{5} + 513050516045 p^{3} T^{6} + 305869032 p^{6} T^{7} + 1553291 p^{9} T^{8} - 566 p^{12} T^{9} + p^{15} T^{10} )^{2}
97 1−4862354T2+13215632715069T4−24078303722833694904T6+ 1 - 4862354 T^{2} + 13215632715069 T^{4} - 24078303722833694904 T^{6} + 32 ⁣⋯ ⁣9832\!\cdots\!98T8− T^{8} - 33 ⁣⋯ ⁣0433\!\cdots\!04T10+ T^{10} + 32 ⁣⋯ ⁣9832\!\cdots\!98p6T12−24078303722833694904p12T14+13215632715069p18T16−4862354p24T18+p30T20 p^{6} T^{12} - 24078303722833694904 p^{12} T^{14} + 13215632715069 p^{18} T^{16} - 4862354 p^{24} T^{18} + p^{30} 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

−3.20012574991461262710978635113, −3.11716804542729319821410206107, −2.97780153356235720578147589868, −2.96830211297804533488749467709, −2.86220030994256645266021020238, −2.81477574390382032654998240856, −2.73857273264401053546307640316, −2.70423261202175805403604164022, −2.36359121162051749389777895123, −2.21412511758739147103541661629, −2.02889000047207762832615659699, −1.82336654075094747969050728901, −1.73646346877050199827174811808, −1.69622047092740210548819898228, −1.67354250906786515053071774960, −1.53606155759569238903463756194, −1.44951430205188373343506235215, −1.11013987304757603036125882511, −0.932431330277150885729181405210, −0.67224859964194900460775949325, −0.62532817631652801605204159546, −0.55404822595746457559784160385, −0.45147084300872390607139243010, −0.39164367490995192215528324954, −0.31907315472650086965856078249, 0.31907315472650086965856078249, 0.39164367490995192215528324954, 0.45147084300872390607139243010, 0.55404822595746457559784160385, 0.62532817631652801605204159546, 0.67224859964194900460775949325, 0.932431330277150885729181405210, 1.11013987304757603036125882511, 1.44951430205188373343506235215, 1.53606155759569238903463756194, 1.67354250906786515053071774960, 1.69622047092740210548819898228, 1.73646346877050199827174811808, 1.82336654075094747969050728901, 2.02889000047207762832615659699, 2.21412511758739147103541661629, 2.36359121162051749389777895123, 2.70423261202175805403604164022, 2.73857273264401053546307640316, 2.81477574390382032654998240856, 2.86220030994256645266021020238, 2.96830211297804533488749467709, 2.97780153356235720578147589868, 3.11716804542729319821410206107, 3.20012574991461262710978635113

Graph of the ZZ-function along the critical line

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