Dirichlet series
| L(s) = 1 | − 3·2-s − 15·3-s + 12·4-s − 25·5-s + 45·6-s − 32·7-s − 29·8-s + 90·9-s + 75·10-s − 43·11-s − 180·12-s + 246·13-s + 96·14-s + 375·15-s + 81·16-s − 124·17-s − 270·18-s − 37·19-s − 300·20-s + 480·21-s + 129·22-s − 77·23-s + 435·24-s + 250·25-s − 738·26-s − 135·27-s − 384·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 2.88·3-s + 3/2·4-s − 2.23·5-s + 3.06·6-s − 1.72·7-s − 1.28·8-s + 10/3·9-s + 2.37·10-s − 1.17·11-s − 4.33·12-s + 5.24·13-s + 1.83·14-s + 6.45·15-s + 1.26·16-s − 1.76·17-s − 3.53·18-s − 0.446·19-s − 3.35·20-s + 4.98·21-s + 1.25·22-s − 0.698·23-s + 3.69·24-s + 2·25-s − 5.56·26-s − 0.962·27-s − 2.59·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 5^{10} \cdot 7^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.32824\times 10^{7}\) |
| Root analytic conductor: | \(2.48901\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 5^{10} \cdot 7^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.004555896636\) |
| \(L(\frac12)\) | \(\approx\) | \(0.004555896636\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p T + p^{2} T^{2} )^{5} \) |
| 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
−5.08703108310779805543292292588, −4.83131862367994517626143551616, −4.75716380527812804619769674177, −4.65236850079770005725902168858, −4.35415658296865154323022544627, −4.19636381504464884056042703456, −4.03070667743362859182128958449, −3.94810715474166964051311053312, −3.81407876482468821749707271274, −3.70739930034245205155414830905, −3.62019457042287640956862342666, −3.31881556637462249107845225845, −2.93094378615670198687058174549, −2.80977033035475969259877980905, −2.74944926665125750190403787528, −2.72021712981761545904342377002, −2.44827642745509074848708831829, −1.85861528504102931978589823275, −1.61147071901546870716630389633, −1.42489597985328168772986461594, −0.979104398831341465789453249294, −0.972257530760754125596569773211, −0.67125472143038036290580231823, −0.38062364287870065718190354804, −0.02708510783300229298331801568, 0.02708510783300229298331801568, 0.38062364287870065718190354804, 0.67125472143038036290580231823, 0.972257530760754125596569773211, 0.979104398831341465789453249294, 1.42489597985328168772986461594, 1.61147071901546870716630389633, 1.85861528504102931978589823275, 2.44827642745509074848708831829, 2.72021712981761545904342377002, 2.74944926665125750190403787528, 2.80977033035475969259877980905, 2.93094378615670198687058174549, 3.31881556637462249107845225845, 3.62019457042287640956862342666, 3.70739930034245205155414830905, 3.81407876482468821749707271274, 3.94810715474166964051311053312, 4.03070667743362859182128958449, 4.19636381504464884056042703456, 4.35415658296865154323022544627, 4.65236850079770005725902168858, 4.75716380527812804619769674177, 4.83131862367994517626143551616, 5.08703108310779805543292292588
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.