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
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{20} \cdot 5^{10} \cdot 7^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.91774\times 10^{12}\) |
| Root analytic conductor: | \(4.31110\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{20} \cdot 5^{10} \cdot 7^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(7.438491794\) |
| \(L(\frac12)\) | \(\approx\) | \(7.438491794\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 - p T + p^{2} T^{2} )^{5} \) | |
| 7 | \( 1 + 32 T + 157 T^{2} - 708 T^{3} - 17741 p T^{4} - 108076 p^{2} T^{5} - 17741 p^{4} T^{6} - 708 p^{6} T^{7} + 157 p^{9} T^{8} + 32 p^{12} T^{9} + p^{15} T^{10} \) | |
| good | 2 | \( 1 - 3 T - 3 T^{2} + p^{4} T^{3} - 3 p T^{4} + 7 p^{2} T^{5} + 43 p^{4} T^{6} - 191 p^{4} T^{7} - 35 p^{3} T^{8} + 297 p^{5} T^{9} + 97 p^{4} T^{10} + 297 p^{8} T^{11} - 35 p^{9} T^{12} - 191 p^{13} T^{13} + 43 p^{16} T^{14} + 7 p^{17} T^{15} - 3 p^{19} T^{16} + p^{25} T^{17} - 3 p^{24} T^{18} - 3 p^{27} T^{19} + p^{30} T^{20} \) |
| 11 | \( 1 - 43 T - 1018 T^{2} + 184703 T^{3} - 5534545 T^{4} - 50464234 T^{5} + 12689409820 T^{6} - 504310395110 T^{7} + 6113678102337 T^{8} + 534835812328359 T^{9} - 31491030462599406 T^{10} + 534835812328359 p^{3} T^{11} + 6113678102337 p^{6} T^{12} - 504310395110 p^{9} T^{13} + 12689409820 p^{12} T^{14} - 50464234 p^{15} T^{15} - 5534545 p^{18} T^{16} + 184703 p^{21} T^{17} - 1018 p^{24} T^{18} - 43 p^{27} T^{19} + p^{30} T^{20} \) | |
| 13 | \( ( 1 - 123 T + 9911 T^{2} - 514334 T^{3} + 26762121 T^{4} - 1198576829 T^{5} + 26762121 p^{3} T^{6} - 514334 p^{6} T^{7} + 9911 p^{9} T^{8} - 123 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 17 | \( 1 - 124 T - 13393 T^{2} + 1317332 T^{3} + 223226567 T^{4} - 13464256864 T^{5} - 1978738682246 T^{6} + 55979698376512 T^{7} + 15589925656986105 T^{8} - 163083275563038492 T^{9} - 82630650088387671747 T^{10} - 163083275563038492 p^{3} T^{11} + 15589925656986105 p^{6} T^{12} + 55979698376512 p^{9} T^{13} - 1978738682246 p^{12} T^{14} - 13464256864 p^{15} T^{15} + 223226567 p^{18} T^{16} + 1317332 p^{21} T^{17} - 13393 p^{24} T^{18} - 124 p^{27} T^{19} + p^{30} T^{20} \) | |
| 19 | \( 1 + 37 T - 21904 T^{2} + 94257 T^{3} + 282384930 T^{4} - 7197449657 T^{5} - 2269051553590 T^{6} + 4234927397735 p T^{7} + 13988764492770637 T^{8} - 261602777288079214 T^{9} - 83432723067784841076 T^{10} - 261602777288079214 p^{3} T^{11} + 13988764492770637 p^{6} T^{12} + 4234927397735 p^{10} T^{13} - 2269051553590 p^{12} T^{14} - 7197449657 p^{15} T^{15} + 282384930 p^{18} T^{16} + 94257 p^{21} T^{17} - 21904 p^{24} T^{18} + 37 p^{27} T^{19} + p^{30} T^{20} \) | |
| 23 | \( 1 - 77 T - 1762 p T^{2} + 3164693 T^{3} + 912830895 T^{4} - 63875502782 T^{5} - 14990398295836 T^{6} + 743770274612358 T^{7} + 208475820275483873 T^{8} - 3628111953906564439 T^{9} - \)\(26\!\cdots\!78\)\( T^{10} - 3628111953906564439 p^{3} T^{11} + 208475820275483873 p^{6} T^{12} + 743770274612358 p^{9} T^{13} - 14990398295836 p^{12} T^{14} - 63875502782 p^{15} T^{15} + 912830895 p^{18} T^{16} + 3164693 p^{21} T^{17} - 1762 p^{25} T^{18} - 77 p^{27} T^{19} + p^{30} T^{20} \) | |
| 29 | \( ( 1 + 360 T + 130325 T^{2} + 27401400 T^{3} + 6029894570 T^{4} + 923796431136 T^{5} + 6029894570 p^{3} T^{6} + 27401400 p^{6} T^{7} + 130325 p^{9} T^{8} + 360 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 31 | \( 1 + 314 T - 25093 T^{2} - 11954934 T^{3} + 865517672 T^{4} + 110001595950 T^{5} - 90893388430871 T^{6} - 6546943301723310 T^{7} + 2772893062489254731 T^{8} + \)\(16\!\cdots\!96\)\( T^{9} - \)\(56\!\cdots\!12\)\( T^{10} + \)\(16\!\cdots\!96\)\( p^{3} T^{11} + 2772893062489254731 p^{6} T^{12} - 6546943301723310 p^{9} T^{13} - 90893388430871 p^{12} T^{14} + 110001595950 p^{15} T^{15} + 865517672 p^{18} T^{16} - 11954934 p^{21} T^{17} - 25093 p^{24} T^{18} + 314 p^{27} T^{19} + p^{30} T^{20} \) | |
| 37 | \( 1 + 225 T - 76286 T^{2} - 2361973 T^{3} + 6089794302 T^{4} - 356399592229 T^{5} - 68864532404144 T^{6} + 56856556722394743 T^{7} - 7573070672827671431 T^{8} - \)\(37\!\cdots\!02\)\( T^{9} + \)\(11\!\cdots\!48\)\( T^{10} - \)\(37\!\cdots\!02\)\( p^{3} T^{11} - 7573070672827671431 p^{6} T^{12} + 56856556722394743 p^{9} T^{13} - 68864532404144 p^{12} T^{14} - 356399592229 p^{15} T^{15} + 6089794302 p^{18} T^{16} - 2361973 p^{21} T^{17} - 76286 p^{24} T^{18} + 225 p^{27} T^{19} + p^{30} T^{20} \) | |
| 41 | \( ( 1 + 341 T + 190073 T^{2} + 39759328 T^{3} + 16717012002 T^{4} + 3155754961314 T^{5} + 16717012002 p^{3} T^{6} + 39759328 p^{6} T^{7} + 190073 p^{9} T^{8} + 341 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 43 | \( ( 1 - 32 T + 218617 T^{2} + 27320644 T^{3} + 18062796109 T^{4} + 4794346174172 T^{5} + 18062796109 p^{3} T^{6} + 27320644 p^{6} T^{7} + 218617 p^{9} T^{8} - 32 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 47 | \( 1 - 25 T - 227610 T^{2} + 73101461 T^{3} + 24026193611 T^{4} - 14842831799166 T^{5} + 815012071632728 T^{6} + 1726119131867110546 T^{7} - \)\(42\!\cdots\!67\)\( T^{8} - \)\(71\!\cdots\!07\)\( T^{9} + \)\(63\!\cdots\!74\)\( T^{10} - \)\(71\!\cdots\!07\)\( p^{3} T^{11} - \)\(42\!\cdots\!67\)\( p^{6} T^{12} + 1726119131867110546 p^{9} T^{13} + 815012071632728 p^{12} T^{14} - 14842831799166 p^{15} T^{15} + 24026193611 p^{18} T^{16} + 73101461 p^{21} T^{17} - 227610 p^{24} T^{18} - 25 p^{27} T^{19} + p^{30} T^{20} \) | |
| 53 | \( 1 + 317 T + 36608 T^{2} - 65490883 T^{3} - 43515633637 T^{4} - 7415247471298 T^{5} + 6175303105639036 T^{6} + 2555693600931772834 T^{7} + \)\(73\!\cdots\!25\)\( T^{8} - \)\(23\!\cdots\!61\)\( T^{9} - \)\(17\!\cdots\!72\)\( T^{10} - \)\(23\!\cdots\!61\)\( p^{3} T^{11} + \)\(73\!\cdots\!25\)\( p^{6} T^{12} + 2555693600931772834 p^{9} T^{13} + 6175303105639036 p^{12} T^{14} - 7415247471298 p^{15} T^{15} - 43515633637 p^{18} T^{16} - 65490883 p^{21} T^{17} + 36608 p^{24} T^{18} + 317 p^{27} T^{19} + p^{30} T^{20} \) | |
| 59 | \( 1 - 676 T - 259179 T^{2} + 22928420 T^{3} + 159667268999 T^{4} + 15850073806032 T^{5} - 23985994336882042 T^{6} - 12685589247886825904 T^{7} + \)\(22\!\cdots\!61\)\( T^{8} + \)\(68\!\cdots\!72\)\( T^{9} + \)\(55\!\cdots\!91\)\( T^{10} + \)\(68\!\cdots\!72\)\( p^{3} T^{11} + \)\(22\!\cdots\!61\)\( p^{6} T^{12} - 12685589247886825904 p^{9} T^{13} - 23985994336882042 p^{12} T^{14} + 15850073806032 p^{15} T^{15} + 159667268999 p^{18} T^{16} + 22928420 p^{21} T^{17} - 259179 p^{24} T^{18} - 676 p^{27} T^{19} + p^{30} T^{20} \) | |
| 61 | \( 1 - 188 T - 431481 T^{2} + 373005124 T^{3} + 29297127399 T^{4} - 121648932752488 T^{5} + 38284746381984330 T^{6} + 12628932742569504968 T^{7} - \)\(11\!\cdots\!39\)\( T^{8} - \)\(36\!\cdots\!84\)\( T^{9} + \)\(15\!\cdots\!05\)\( T^{10} - \)\(36\!\cdots\!84\)\( p^{3} T^{11} - \)\(11\!\cdots\!39\)\( p^{6} T^{12} + 12628932742569504968 p^{9} T^{13} + 38284746381984330 p^{12} T^{14} - 121648932752488 p^{15} T^{15} + 29297127399 p^{18} T^{16} + 373005124 p^{21} T^{17} - 431481 p^{24} T^{18} - 188 p^{27} T^{19} + p^{30} T^{20} \) | |
| 67 | \( 1 - 1776 T + 1302831 T^{2} - 795082424 T^{3} + 523494226340 T^{4} - 209497188661456 T^{5} + 62623175581904481 T^{6} - 31980651664564177744 T^{7} + \)\(13\!\cdots\!23\)\( T^{8} + \)\(42\!\cdots\!28\)\( T^{9} - \)\(47\!\cdots\!52\)\( T^{10} + \)\(42\!\cdots\!28\)\( p^{3} T^{11} + \)\(13\!\cdots\!23\)\( p^{6} T^{12} - 31980651664564177744 p^{9} T^{13} + 62623175581904481 p^{12} T^{14} - 209497188661456 p^{15} T^{15} + 523494226340 p^{18} T^{16} - 795082424 p^{21} T^{17} + 1302831 p^{24} T^{18} - 1776 p^{27} T^{19} + p^{30} T^{20} \) | |
| 71 | \( ( 1 - 6 T + 797207 T^{2} + 67070712 T^{3} + 425455729610 T^{4} + 37303525976244 T^{5} + 425455729610 p^{3} T^{6} + 67070712 p^{6} T^{7} + 797207 p^{9} T^{8} - 6 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 73 | \( 1 + 2006 T + 2213445 T^{2} + 2068190842 T^{3} + 22153879952 p T^{4} + 980484252789530 T^{5} + 644957368099789599 T^{6} + \)\(49\!\cdots\!02\)\( T^{7} + \)\(36\!\cdots\!03\)\( T^{8} + \)\(27\!\cdots\!32\)\( T^{9} + \)\(19\!\cdots\!20\)\( T^{10} + \)\(27\!\cdots\!32\)\( p^{3} T^{11} + \)\(36\!\cdots\!03\)\( p^{6} T^{12} + \)\(49\!\cdots\!02\)\( p^{9} T^{13} + 644957368099789599 p^{12} T^{14} + 980484252789530 p^{15} T^{15} + 22153879952 p^{19} T^{16} + 2068190842 p^{21} T^{17} + 2213445 p^{24} T^{18} + 2006 p^{27} T^{19} + p^{30} T^{20} \) | |
| 79 | \( 1 + 200 T - 1905761 T^{2} - 521788304 T^{3} + 2022855950280 T^{4} + 568166623235112 T^{5} - 1467320191398068699 T^{6} - \)\(32\!\cdots\!56\)\( T^{7} + \)\(10\!\cdots\!57\)\( p T^{8} + \)\(73\!\cdots\!56\)\( T^{9} - \)\(42\!\cdots\!44\)\( T^{10} + \)\(73\!\cdots\!56\)\( p^{3} T^{11} + \)\(10\!\cdots\!57\)\( p^{7} T^{12} - \)\(32\!\cdots\!56\)\( p^{9} T^{13} - 1467320191398068699 p^{12} T^{14} + 568166623235112 p^{15} T^{15} + 2022855950280 p^{18} T^{16} - 521788304 p^{21} T^{17} - 1905761 p^{24} T^{18} + 200 p^{27} T^{19} + p^{30} T^{20} \) | |
| 83 | \( ( 1 - 4 p T + 1330763 T^{2} - 251179960 T^{3} + 1196075575498 T^{4} - 287374399354632 T^{5} + 1196075575498 p^{3} T^{6} - 251179960 p^{6} T^{7} + 1330763 p^{9} T^{8} - 4 p^{13} T^{9} + p^{15} T^{10} )^{2} \) | |
| 89 | \( 1 - 894 T - 1501253 T^{2} + 1203783426 T^{3} + 1217576227911 T^{4} - 488918737763772 T^{5} - 941190453323848894 T^{6} - \)\(20\!\cdots\!20\)\( T^{7} + \)\(93\!\cdots\!81\)\( T^{8} + \)\(16\!\cdots\!42\)\( T^{9} - \)\(80\!\cdots\!11\)\( T^{10} + \)\(16\!\cdots\!42\)\( p^{3} T^{11} + \)\(93\!\cdots\!81\)\( p^{6} T^{12} - \)\(20\!\cdots\!20\)\( p^{9} T^{13} - 941190453323848894 p^{12} T^{14} - 488918737763772 p^{15} T^{15} + 1217576227911 p^{18} T^{16} + 1203783426 p^{21} T^{17} - 1501253 p^{24} T^{18} - 894 p^{27} T^{19} + p^{30} T^{20} \) | |
| 97 | \( ( 1 + 576 T + 2417013 T^{2} + 1510894720 T^{3} + 3259586132218 T^{4} + 2005403804499072 T^{5} + 3259586132218 p^{3} T^{6} + 1510894720 p^{6} T^{7} + 2417013 p^{9} T^{8} + 576 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
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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
−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 $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.