↑

Properties

Label 20-315e10-1.1-c3e10-0-1
Degree 2020
Conductor 9.618×10249.618\times 10^{24}
Sign 11
Analytic cond. 4.91774×10124.91774\times 10^{12}
Root an. cond. 4.311104.31110
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 12·4-s + 25·5-s − 32·7-s + 29·8-s + 75·10-s + 43·11-s + 246·13-s − 96·14-s + 81·16-s + 124·17-s − 37·19-s + 300·20-s + 129·22-s + 77·23-s + 250·25-s + 738·26-s − 384·28-s − 720·29-s − 314·31-s + 128·32-s + 372·34-s − 800·35-s − 225·37-s − 111·38-s + 725·40-s − 682·41-s + ⋯
L(s)  = 1  + 1.06·2-s + 3/2·4-s + 2.23·5-s − 1.72·7-s + 1.28·8-s + 2.37·10-s + 1.17·11-s + 5.24·13-s − 1.83·14-s + 1.26·16-s + 1.76·17-s − 0.446·19-s + 3.35·20-s + 1.25·22-s + 0.698·23-s + 2·25-s + 5.56·26-s − 2.59·28-s − 4.61·29-s − 1.81·31-s + 0.707·32-s + 1.87·34-s − 3.86·35-s − 0.999·37-s − 0.473·38-s + 2.86·40-s − 2.59·41-s + ⋯

Functional equation

Λ(s)=((320⋅510⋅710)s/2 ΓC(s)10 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((320⋅510⋅710)s/2 ΓC(s+3/2)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2020
Conductor: 320⋅510⋅7103^{20} \cdot 5^{10} \cdot 7^{10}
Sign: 11
Analytic conductor: 4.91774×10124.91774\times 10^{12}
Root analytic conductor: 4.311104.31110
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (20, 320⋅510⋅710, ( :[3/2]10), 1)(20,\ 3^{20} \cdot 5^{10} \cdot 7^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 7.4384917947.438491794
L(12)L(\frac12) ≈\approx 7.4384917947.438491794
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)
bad3 1 1
5 (1−pT+p2T2)5 ( 1 - p T + p^{2} T^{2} )^{5}
7 1+32T+157T2−708T3−17741pT4−108076p2T5−17741p4T6−708p6T7+157p9T8+32p12T9+p15T10 1 + 32 T + 157 T^{2} - 708 T^{3} - 17741 p T^{4} - 108076 p^{2} T^{5} - 17741 p^{4} T^{6} - 708 p^{6} T^{7} + 157 p^{9} T^{8} + 32 p^{12} T^{9} + p^{15} T^{10}
good2 1−3T−3T2+p4T3−3pT4+7p2T5+43p4T6−191p4T7−35p3T8+297p5T9+97p4T10+297p8T11−35p9T12−191p13T13+43p16T14+7p17T15−3p19T16+p25T17−3p24T18−3p27T19+p30T20 1 - 3 T - 3 T^{2} + p^{4} T^{3} - 3 p T^{4} + 7 p^{2} T^{5} + 43 p^{4} T^{6} - 191 p^{4} T^{7} - 35 p^{3} T^{8} + 297 p^{5} T^{9} + 97 p^{4} T^{10} + 297 p^{8} T^{11} - 35 p^{9} T^{12} - 191 p^{13} T^{13} + 43 p^{16} T^{14} + 7 p^{17} T^{15} - 3 p^{19} T^{16} + p^{25} T^{17} - 3 p^{24} T^{18} - 3 p^{27} T^{19} + p^{30} T^{20}
11 1−43T−1018T2+184703T3−5534545T4−50464234T5+12689409820T6−504310395110T7+6113678102337T8+534835812328359T9−31491030462599406T10+534835812328359p3T11+6113678102337p6T12−504310395110p9T13+12689409820p12T14−50464234p15T15−5534545p18T16+184703p21T17−1018p24T18−43p27T19+p30T20 1 - 43 T - 1018 T^{2} + 184703 T^{3} - 5534545 T^{4} - 50464234 T^{5} + 12689409820 T^{6} - 504310395110 T^{7} + 6113678102337 T^{8} + 534835812328359 T^{9} - 31491030462599406 T^{10} + 534835812328359 p^{3} T^{11} + 6113678102337 p^{6} T^{12} - 504310395110 p^{9} T^{13} + 12689409820 p^{12} T^{14} - 50464234 p^{15} T^{15} - 5534545 p^{18} T^{16} + 184703 p^{21} T^{17} - 1018 p^{24} T^{18} - 43 p^{27} T^{19} + p^{30} T^{20}
13 (1−123T+9911T2−514334T3+26762121T4−1198576829T5+26762121p3T6−514334p6T7+9911p9T8−123p12T9+p15T10)2 ( 1 - 123 T + 9911 T^{2} - 514334 T^{3} + 26762121 T^{4} - 1198576829 T^{5} + 26762121 p^{3} T^{6} - 514334 p^{6} T^{7} + 9911 p^{9} T^{8} - 123 p^{12} T^{9} + p^{15} T^{10} )^{2}
17 1−124T−13393T2+1317332T3+223226567T4−13464256864T5−1978738682246T6+55979698376512T7+15589925656986105T8−163083275563038492T9−82630650088387671747T10−163083275563038492p3T11+15589925656986105p6T12+55979698376512p9T13−1978738682246p12T14−13464256864p15T15+223226567p18T16+1317332p21T17−13393p24T18−124p27T19+p30T20 1 - 124 T - 13393 T^{2} + 1317332 T^{3} + 223226567 T^{4} - 13464256864 T^{5} - 1978738682246 T^{6} + 55979698376512 T^{7} + 15589925656986105 T^{8} - 163083275563038492 T^{9} - 82630650088387671747 T^{10} - 163083275563038492 p^{3} T^{11} + 15589925656986105 p^{6} T^{12} + 55979698376512 p^{9} T^{13} - 1978738682246 p^{12} T^{14} - 13464256864 p^{15} T^{15} + 223226567 p^{18} T^{16} + 1317332 p^{21} T^{17} - 13393 p^{24} T^{18} - 124 p^{27} T^{19} + p^{30} T^{20}
19 1+37T−21904T2+94257T3+282384930T4−7197449657T5−2269051553590T6+4234927397735pT7+13988764492770637T8−261602777288079214T9−83432723067784841076T10−261602777288079214p3T11+13988764492770637p6T12+4234927397735p10T13−2269051553590p12T14−7197449657p15T15+282384930p18T16+94257p21T17−21904p24T18+37p27T19+p30T20 1 + 37 T - 21904 T^{2} + 94257 T^{3} + 282384930 T^{4} - 7197449657 T^{5} - 2269051553590 T^{6} + 4234927397735 p T^{7} + 13988764492770637 T^{8} - 261602777288079214 T^{9} - 83432723067784841076 T^{10} - 261602777288079214 p^{3} T^{11} + 13988764492770637 p^{6} T^{12} + 4234927397735 p^{10} T^{13} - 2269051553590 p^{12} T^{14} - 7197449657 p^{15} T^{15} + 282384930 p^{18} T^{16} + 94257 p^{21} T^{17} - 21904 p^{24} T^{18} + 37 p^{27} T^{19} + p^{30} T^{20}
23 1−77T−1762pT2+3164693T3+912830895T4−63875502782T5−14990398295836T6+743770274612358T7+208475820275483873T8−3628111953906564439T9− 1 - 77 T - 1762 p T^{2} + 3164693 T^{3} + 912830895 T^{4} - 63875502782 T^{5} - 14990398295836 T^{6} + 743770274612358 T^{7} + 208475820275483873 T^{8} - 3628111953906564439 T^{9} - 26 ⁣⋯ ⁣7826\!\cdots\!78T10−3628111953906564439p3T11+208475820275483873p6T12+743770274612358p9T13−14990398295836p12T14−63875502782p15T15+912830895p18T16+3164693p21T17−1762p25T18−77p27T19+p30T20 T^{10} - 3628111953906564439 p^{3} T^{11} + 208475820275483873 p^{6} T^{12} + 743770274612358 p^{9} T^{13} - 14990398295836 p^{12} T^{14} - 63875502782 p^{15} T^{15} + 912830895 p^{18} T^{16} + 3164693 p^{21} T^{17} - 1762 p^{25} T^{18} - 77 p^{27} T^{19} + p^{30} T^{20}
29 (1+360T+130325T2+27401400T3+6029894570T4+923796431136T5+6029894570p3T6+27401400p6T7+130325p9T8+360p12T9+p15T10)2 ( 1 + 360 T + 130325 T^{2} + 27401400 T^{3} + 6029894570 T^{4} + 923796431136 T^{5} + 6029894570 p^{3} T^{6} + 27401400 p^{6} T^{7} + 130325 p^{9} T^{8} + 360 p^{12} T^{9} + p^{15} T^{10} )^{2}
31 1+314T−25093T2−11954934T3+865517672T4+110001595950T5−90893388430871T6−6546943301723310T7+2772893062489254731T8+ 1 + 314 T - 25093 T^{2} - 11954934 T^{3} + 865517672 T^{4} + 110001595950 T^{5} - 90893388430871 T^{6} - 6546943301723310 T^{7} + 2772893062489254731 T^{8} + 16 ⁣⋯ ⁣9616\!\cdots\!96T9− T^{9} - 56 ⁣⋯ ⁣1256\!\cdots\!12T10+ T^{10} + 16 ⁣⋯ ⁣9616\!\cdots\!96p3T11+2772893062489254731p6T12−6546943301723310p9T13−90893388430871p12T14+110001595950p15T15+865517672p18T16−11954934p21T17−25093p24T18+314p27T19+p30T20 p^{3} T^{11} + 2772893062489254731 p^{6} T^{12} - 6546943301723310 p^{9} T^{13} - 90893388430871 p^{12} T^{14} + 110001595950 p^{15} T^{15} + 865517672 p^{18} T^{16} - 11954934 p^{21} T^{17} - 25093 p^{24} T^{18} + 314 p^{27} T^{19} + p^{30} T^{20}
37 1+225T−76286T2−2361973T3+6089794302T4−356399592229T5−68864532404144T6+56856556722394743T7−7573070672827671431T8− 1 + 225 T - 76286 T^{2} - 2361973 T^{3} + 6089794302 T^{4} - 356399592229 T^{5} - 68864532404144 T^{6} + 56856556722394743 T^{7} - 7573070672827671431 T^{8} - 37 ⁣⋯ ⁣0237\!\cdots\!02T9+ T^{9} + 11 ⁣⋯ ⁣4811\!\cdots\!48T10− T^{10} - 37 ⁣⋯ ⁣0237\!\cdots\!02p3T11−7573070672827671431p6T12+56856556722394743p9T13−68864532404144p12T14−356399592229p15T15+6089794302p18T16−2361973p21T17−76286p24T18+225p27T19+p30T20 p^{3} T^{11} - 7573070672827671431 p^{6} T^{12} + 56856556722394743 p^{9} T^{13} - 68864532404144 p^{12} T^{14} - 356399592229 p^{15} T^{15} + 6089794302 p^{18} T^{16} - 2361973 p^{21} T^{17} - 76286 p^{24} T^{18} + 225 p^{27} T^{19} + p^{30} T^{20}
41 (1+341T+190073T2+39759328T3+16717012002T4+3155754961314T5+16717012002p3T6+39759328p6T7+190073p9T8+341p12T9+p15T10)2 ( 1 + 341 T + 190073 T^{2} + 39759328 T^{3} + 16717012002 T^{4} + 3155754961314 T^{5} + 16717012002 p^{3} T^{6} + 39759328 p^{6} T^{7} + 190073 p^{9} T^{8} + 341 p^{12} T^{9} + p^{15} T^{10} )^{2}
43 (1−32T+218617T2+27320644T3+18062796109T4+4794346174172T5+18062796109p3T6+27320644p6T7+218617p9T8−32p12T9+p15T10)2 ( 1 - 32 T + 218617 T^{2} + 27320644 T^{3} + 18062796109 T^{4} + 4794346174172 T^{5} + 18062796109 p^{3} T^{6} + 27320644 p^{6} T^{7} + 218617 p^{9} T^{8} - 32 p^{12} T^{9} + p^{15} T^{10} )^{2}
47 1−25T−227610T2+73101461T3+24026193611T4−14842831799166T5+815012071632728T6+1726119131867110546T7− 1 - 25 T - 227610 T^{2} + 73101461 T^{3} + 24026193611 T^{4} - 14842831799166 T^{5} + 815012071632728 T^{6} + 1726119131867110546 T^{7} - 42 ⁣⋯ ⁣6742\!\cdots\!67T8− T^{8} - 71 ⁣⋯ ⁣0771\!\cdots\!07T9+ T^{9} + 63 ⁣⋯ ⁣7463\!\cdots\!74T10− T^{10} - 71 ⁣⋯ ⁣0771\!\cdots\!07p3T11− p^{3} T^{11} - 42 ⁣⋯ ⁣6742\!\cdots\!67p6T12+1726119131867110546p9T13+815012071632728p12T14−14842831799166p15T15+24026193611p18T16+73101461p21T17−227610p24T18−25p27T19+p30T20 p^{6} T^{12} + 1726119131867110546 p^{9} T^{13} + 815012071632728 p^{12} T^{14} - 14842831799166 p^{15} T^{15} + 24026193611 p^{18} T^{16} + 73101461 p^{21} T^{17} - 227610 p^{24} T^{18} - 25 p^{27} T^{19} + p^{30} T^{20}
53 1+317T+36608T2−65490883T3−43515633637T4−7415247471298T5+6175303105639036T6+2555693600931772834T7+ 1 + 317 T + 36608 T^{2} - 65490883 T^{3} - 43515633637 T^{4} - 7415247471298 T^{5} + 6175303105639036 T^{6} + 2555693600931772834 T^{7} + 73 ⁣⋯ ⁣2573\!\cdots\!25T8− T^{8} - 23 ⁣⋯ ⁣6123\!\cdots\!61T9− T^{9} - 17 ⁣⋯ ⁣7217\!\cdots\!72T10− T^{10} - 23 ⁣⋯ ⁣6123\!\cdots\!61p3T11+ p^{3} T^{11} + 73 ⁣⋯ ⁣2573\!\cdots\!25p6T12+2555693600931772834p9T13+6175303105639036p12T14−7415247471298p15T15−43515633637p18T16−65490883p21T17+36608p24T18+317p27T19+p30T20 p^{6} T^{12} + 2555693600931772834 p^{9} T^{13} + 6175303105639036 p^{12} T^{14} - 7415247471298 p^{15} T^{15} - 43515633637 p^{18} T^{16} - 65490883 p^{21} T^{17} + 36608 p^{24} T^{18} + 317 p^{27} T^{19} + p^{30} T^{20}
59 1−676T−259179T2+22928420T3+159667268999T4+15850073806032T5−23985994336882042T6−12685589247886825904T7+ 1 - 676 T - 259179 T^{2} + 22928420 T^{3} + 159667268999 T^{4} + 15850073806032 T^{5} - 23985994336882042 T^{6} - 12685589247886825904 T^{7} + 22 ⁣⋯ ⁣6122\!\cdots\!61T8+ T^{8} + 68 ⁣⋯ ⁣7268\!\cdots\!72T9+ T^{9} + 55 ⁣⋯ ⁣9155\!\cdots\!91T10+ T^{10} + 68 ⁣⋯ ⁣7268\!\cdots\!72p3T11+ p^{3} T^{11} + 22 ⁣⋯ ⁣6122\!\cdots\!61p6T12−12685589247886825904p9T13−23985994336882042p12T14+15850073806032p15T15+159667268999p18T16+22928420p21T17−259179p24T18−676p27T19+p30T20 p^{6} T^{12} - 12685589247886825904 p^{9} T^{13} - 23985994336882042 p^{12} T^{14} + 15850073806032 p^{15} T^{15} + 159667268999 p^{18} T^{16} + 22928420 p^{21} T^{17} - 259179 p^{24} T^{18} - 676 p^{27} T^{19} + p^{30} T^{20}
61 1−188T−431481T2+373005124T3+29297127399T4−121648932752488T5+38284746381984330T6+12628932742569504968T7− 1 - 188 T - 431481 T^{2} + 373005124 T^{3} + 29297127399 T^{4} - 121648932752488 T^{5} + 38284746381984330 T^{6} + 12628932742569504968 T^{7} - 11 ⁣⋯ ⁣3911\!\cdots\!39T8− T^{8} - 36 ⁣⋯ ⁣8436\!\cdots\!84T9+ T^{9} + 15 ⁣⋯ ⁣0515\!\cdots\!05T10− T^{10} - 36 ⁣⋯ ⁣8436\!\cdots\!84p3T11− p^{3} T^{11} - 11 ⁣⋯ ⁣3911\!\cdots\!39p6T12+12628932742569504968p9T13+38284746381984330p12T14−121648932752488p15T15+29297127399p18T16+373005124p21T17−431481p24T18−188p27T19+p30T20 p^{6} T^{12} + 12628932742569504968 p^{9} T^{13} + 38284746381984330 p^{12} T^{14} - 121648932752488 p^{15} T^{15} + 29297127399 p^{18} T^{16} + 373005124 p^{21} T^{17} - 431481 p^{24} T^{18} - 188 p^{27} T^{19} + p^{30} T^{20}
67 1−1776T+1302831T2−795082424T3+523494226340T4−209497188661456T5+62623175581904481T6−31980651664564177744T7+ 1 - 1776 T + 1302831 T^{2} - 795082424 T^{3} + 523494226340 T^{4} - 209497188661456 T^{5} + 62623175581904481 T^{6} - 31980651664564177744 T^{7} + 13 ⁣⋯ ⁣2313\!\cdots\!23T8+ T^{8} + 42 ⁣⋯ ⁣2842\!\cdots\!28T9− T^{9} - 47 ⁣⋯ ⁣5247\!\cdots\!52T10+ T^{10} + 42 ⁣⋯ ⁣2842\!\cdots\!28p3T11+ p^{3} T^{11} + 13 ⁣⋯ ⁣2313\!\cdots\!23p6T12−31980651664564177744p9T13+62623175581904481p12T14−209497188661456p15T15+523494226340p18T16−795082424p21T17+1302831p24T18−1776p27T19+p30T20 p^{6} T^{12} - 31980651664564177744 p^{9} T^{13} + 62623175581904481 p^{12} T^{14} - 209497188661456 p^{15} T^{15} + 523494226340 p^{18} T^{16} - 795082424 p^{21} T^{17} + 1302831 p^{24} T^{18} - 1776 p^{27} T^{19} + p^{30} T^{20}
71 (1−6T+797207T2+67070712T3+425455729610T4+37303525976244T5+425455729610p3T6+67070712p6T7+797207p9T8−6p12T9+p15T10)2 ( 1 - 6 T + 797207 T^{2} + 67070712 T^{3} + 425455729610 T^{4} + 37303525976244 T^{5} + 425455729610 p^{3} T^{6} + 67070712 p^{6} T^{7} + 797207 p^{9} T^{8} - 6 p^{12} T^{9} + p^{15} T^{10} )^{2}
73 1+2006T+2213445T2+2068190842T3+22153879952pT4+980484252789530T5+644957368099789599T6+ 1 + 2006 T + 2213445 T^{2} + 2068190842 T^{3} + 22153879952 p T^{4} + 980484252789530 T^{5} + 644957368099789599 T^{6} + 49 ⁣⋯ ⁣0249\!\cdots\!02T7+ T^{7} + 36 ⁣⋯ ⁣0336\!\cdots\!03T8+ T^{8} + 27 ⁣⋯ ⁣3227\!\cdots\!32T9+ T^{9} + 19 ⁣⋯ ⁣2019\!\cdots\!20T10+ T^{10} + 27 ⁣⋯ ⁣3227\!\cdots\!32p3T11+ p^{3} T^{11} + 36 ⁣⋯ ⁣0336\!\cdots\!03p6T12+ p^{6} T^{12} + 49 ⁣⋯ ⁣0249\!\cdots\!02p9T13+644957368099789599p12T14+980484252789530p15T15+22153879952p19T16+2068190842p21T17+2213445p24T18+2006p27T19+p30T20 p^{9} T^{13} + 644957368099789599 p^{12} T^{14} + 980484252789530 p^{15} T^{15} + 22153879952 p^{19} T^{16} + 2068190842 p^{21} T^{17} + 2213445 p^{24} T^{18} + 2006 p^{27} T^{19} + p^{30} T^{20}
79 1+200T−1905761T2−521788304T3+2022855950280T4+568166623235112T5−1467320191398068699T6− 1 + 200 T - 1905761 T^{2} - 521788304 T^{3} + 2022855950280 T^{4} + 568166623235112 T^{5} - 1467320191398068699 T^{6} - 32 ⁣⋯ ⁣5632\!\cdots\!56T7+ T^{7} + 10 ⁣⋯ ⁣5710\!\cdots\!57pT8+ p T^{8} + 73 ⁣⋯ ⁣5673\!\cdots\!56T9− T^{9} - 42 ⁣⋯ ⁣4442\!\cdots\!44T10+ T^{10} + 73 ⁣⋯ ⁣5673\!\cdots\!56p3T11+ p^{3} T^{11} + 10 ⁣⋯ ⁣5710\!\cdots\!57p7T12− p^{7} T^{12} - 32 ⁣⋯ ⁣5632\!\cdots\!56p9T13−1467320191398068699p12T14+568166623235112p15T15+2022855950280p18T16−521788304p21T17−1905761p24T18+200p27T19+p30T20 p^{9} T^{13} - 1467320191398068699 p^{12} T^{14} + 568166623235112 p^{15} T^{15} + 2022855950280 p^{18} T^{16} - 521788304 p^{21} T^{17} - 1905761 p^{24} T^{18} + 200 p^{27} T^{19} + p^{30} T^{20}
83 (1−4pT+1330763T2−251179960T3+1196075575498T4−287374399354632T5+1196075575498p3T6−251179960p6T7+1330763p9T8−4p13T9+p15T10)2 ( 1 - 4 p T + 1330763 T^{2} - 251179960 T^{3} + 1196075575498 T^{4} - 287374399354632 T^{5} + 1196075575498 p^{3} T^{6} - 251179960 p^{6} T^{7} + 1330763 p^{9} T^{8} - 4 p^{13} T^{9} + p^{15} T^{10} )^{2}
89 1−894T−1501253T2+1203783426T3+1217576227911T4−488918737763772T5−941190453323848894T6− 1 - 894 T - 1501253 T^{2} + 1203783426 T^{3} + 1217576227911 T^{4} - 488918737763772 T^{5} - 941190453323848894 T^{6} - 20 ⁣⋯ ⁣2020\!\cdots\!20T7+ T^{7} + 93 ⁣⋯ ⁣8193\!\cdots\!81T8+ T^{8} + 16 ⁣⋯ ⁣4216\!\cdots\!42T9− T^{9} - 80 ⁣⋯ ⁣1180\!\cdots\!11T10+ T^{10} + 16 ⁣⋯ ⁣4216\!\cdots\!42p3T11+ p^{3} T^{11} + 93 ⁣⋯ ⁣8193\!\cdots\!81p6T12− p^{6} T^{12} - 20 ⁣⋯ ⁣2020\!\cdots\!20p9T13−941190453323848894p12T14−488918737763772p15T15+1217576227911p18T16+1203783426p21T17−1501253p24T18−894p27T19+p30T20 p^{9} T^{13} - 941190453323848894 p^{12} T^{14} - 488918737763772 p^{15} T^{15} + 1217576227911 p^{18} T^{16} + 1203783426 p^{21} T^{17} - 1501253 p^{24} T^{18} - 894 p^{27} T^{19} + p^{30} T^{20}
97 (1+576T+2417013T2+1510894720T3+3259586132218T4+2005403804499072T5+3259586132218p3T6+1510894720p6T7+2417013p9T8+576p12T9+p15T10)2 ( 1 + 576 T + 2417013 T^{2} + 1510894720 T^{3} + 3259586132218 T^{4} + 2005403804499072 T^{5} + 3259586132218 p^{3} T^{6} + 1510894720 p^{6} T^{7} + 2417013 p^{9} T^{8} + 576 p^{12} T^{9} + p^{15} T^{10} )^{2}
show more
show less
   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

−3.87898304460970020092644034689, −3.76097186027411812682173719988, −3.65565385596705709332454895149, −3.60945312069074178779110713126, −3.46351798864721185164565591722, −3.31219610970595203910339479328, −3.26976692623893243098452479620, −3.26433742236882016589133446504, −3.05616728402439692331794970803, −2.83082883737908951855221857088, −2.56073137857304299641315079139, −2.35261879495777723366847468106, −2.26513712783988032839230661617, −2.18523721576198602318542350191, −1.93648056786226328864048893963, −1.78208781863390981423076064624, −1.64838395338339172609851575023, −1.53251230945532898159516175568, −1.40894324555841698362598519748, −1.16125601582875576190518848833, −1.13648663563468330559242619539, −1.08311777802858820367562959432, −0.66299987889855824141617531863, −0.28791064512092849066585180742, −0.095864767232451704997591039998, 0.095864767232451704997591039998, 0.28791064512092849066585180742, 0.66299987889855824141617531863, 1.08311777802858820367562959432, 1.13648663563468330559242619539, 1.16125601582875576190518848833, 1.40894324555841698362598519748, 1.53251230945532898159516175568, 1.64838395338339172609851575023, 1.78208781863390981423076064624, 1.93648056786226328864048893963, 2.18523721576198602318542350191, 2.26513712783988032839230661617, 2.35261879495777723366847468106, 2.56073137857304299641315079139, 2.83082883737908951855221857088, 3.05616728402439692331794970803, 3.26433742236882016589133446504, 3.26976692623893243098452479620, 3.31219610970595203910339479328, 3.46351798864721185164565591722, 3.60945312069074178779110713126, 3.65565385596705709332454895149, 3.76097186027411812682173719988, 3.87898304460970020092644034689

Graph of the ZZ-function along the critical line

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