Dirichlet series
| L(s) = 1 | + 3-s − 4·7-s − 11-s − 6·13-s − 6·17-s + 18·19-s − 4·21-s + 4·23-s + 13·25-s − 27-s + 4·29-s − 8·31-s − 33-s + 20·37-s − 6·39-s − 5·41-s − 13·43-s − 6·47-s + 27·49-s − 6·51-s + 18·57-s − 13·59-s − 10·61-s − 17·67-s + 4·69-s + 8·71-s − 34·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s − 0.301·11-s − 1.66·13-s − 1.45·17-s + 4.12·19-s − 0.872·21-s + 0.834·23-s + 13/5·25-s − 0.192·27-s + 0.742·29-s − 1.43·31-s − 0.174·33-s + 3.28·37-s − 0.960·39-s − 0.780·41-s − 1.98·43-s − 0.875·47-s + 27/7·49-s − 0.840·51-s + 2.38·57-s − 1.69·59-s − 1.28·61-s − 2.07·67-s + 0.481·69-s + 0.949·71-s − 3.97·73-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{70} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{70} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{70} \cdot 3^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.33806\times 10^{9}\) |
| Root analytic conductor: | \(3.03294\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{70} \cdot 3^{20} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(12.65298110\) |
| \(L(\frac12)\) | \(\approx\) | \(12.65298110\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T + T^{2} - 2 p^{2} T^{5} + p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 5 | \( 1 - 13 T^{2} - 4 T^{3} + 72 T^{4} + 52 T^{5} - 287 T^{6} - 432 T^{7} + 1427 T^{8} + 1232 T^{9} - 7344 T^{10} + 1232 p T^{11} + 1427 p^{2} T^{12} - 432 p^{3} T^{13} - 287 p^{4} T^{14} + 52 p^{5} T^{15} + 72 p^{6} T^{16} - 4 p^{7} T^{17} - 13 p^{8} T^{18} + p^{10} T^{20} \) |
| 7 | \( 1 + 4 T - 11 T^{2} - 12 p T^{3} + 24 T^{4} + 978 T^{5} + 1227 T^{6} - 7008 T^{7} - 20841 T^{8} + 19654 T^{9} + 182116 T^{10} + 19654 p T^{11} - 20841 p^{2} T^{12} - 7008 p^{3} T^{13} + 1227 p^{4} T^{14} + 978 p^{5} T^{15} + 24 p^{6} T^{16} - 12 p^{8} T^{17} - 11 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + T - 34 T^{2} + 47 T^{3} + 692 T^{4} - 1631 T^{5} - 6628 T^{6} + 25979 T^{7} + 32911 T^{8} - 11076 p T^{9} + 2820 T^{10} - 11076 p^{2} T^{11} + 32911 p^{2} T^{12} + 25979 p^{3} T^{13} - 6628 p^{4} T^{14} - 1631 p^{5} T^{15} + 692 p^{6} T^{16} + 47 p^{7} T^{17} - 34 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 6 T - 29 T^{2} - 158 T^{3} + 912 T^{4} + 2926 T^{5} - 20879 T^{6} - 34926 T^{7} + 378667 T^{8} + 219368 T^{9} - 5274240 T^{10} + 219368 p T^{11} + 378667 p^{2} T^{12} - 34926 p^{3} T^{13} - 20879 p^{4} T^{14} + 2926 p^{5} T^{15} + 912 p^{6} T^{16} - 158 p^{7} T^{17} - 29 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( ( 1 + 3 T + 61 T^{2} + 124 T^{3} + 98 p T^{4} + 2590 T^{5} + 98 p^{2} T^{6} + 124 p^{2} T^{7} + 61 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 19 | \( ( 1 - 9 T + 5 p T^{2} - 580 T^{3} + 3562 T^{4} - 15686 T^{5} + 3562 p T^{6} - 580 p^{2} T^{7} + 5 p^{4} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 23 | \( 1 - 4 T - 67 T^{2} + 448 T^{3} + 2204 T^{4} - 21046 T^{5} - 18229 T^{6} + 557656 T^{7} - 38827 p T^{8} - 5599446 T^{9} + 38685828 T^{10} - 5599446 p T^{11} - 38827 p^{3} T^{12} + 557656 p^{3} T^{13} - 18229 p^{4} T^{14} - 21046 p^{5} T^{15} + 2204 p^{6} T^{16} + 448 p^{7} T^{17} - 67 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 4 T - 49 T^{2} + 16 p T^{3} + 28 T^{4} - 15992 T^{5} + 53985 T^{6} + 260556 T^{7} - 2006961 T^{8} - 2487664 T^{9} + 53816488 T^{10} - 2487664 p T^{11} - 2006961 p^{2} T^{12} + 260556 p^{3} T^{13} + 53985 p^{4} T^{14} - 15992 p^{5} T^{15} + 28 p^{6} T^{16} + 16 p^{8} T^{17} - 49 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 8 T - 23 T^{2} - 436 T^{3} - 1532 T^{4} + 566 T^{5} + 10095 T^{6} + 4740 T^{7} + 827475 T^{8} + 4805962 T^{9} + 6955412 T^{10} + 4805962 p T^{11} + 827475 p^{2} T^{12} + 4740 p^{3} T^{13} + 10095 p^{4} T^{14} + 566 p^{5} T^{15} - 1532 p^{6} T^{16} - 436 p^{7} T^{17} - 23 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( ( 1 - 10 T + 117 T^{2} - 684 T^{3} + 4914 T^{4} - 23580 T^{5} + 4914 p T^{6} - 684 p^{2} T^{7} + 117 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 41 | \( 1 + 5 T - 34 T^{2} - 229 T^{3} - 1526 T^{4} + 1039 T^{5} + 40920 T^{6} - 494097 T^{7} - 845715 T^{8} + 22003682 T^{9} + 126104644 T^{10} + 22003682 p T^{11} - 845715 p^{2} T^{12} - 494097 p^{3} T^{13} + 40920 p^{4} T^{14} + 1039 p^{5} T^{15} - 1526 p^{6} T^{16} - 229 p^{7} T^{17} - 34 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 13 T + 4 T^{2} - 239 T^{3} + 2470 T^{4} + 11233 T^{5} - 29328 T^{6} + 171807 T^{7} - 1680075 T^{8} + 8058734 T^{9} + 347494688 T^{10} + 8058734 p T^{11} - 1680075 p^{2} T^{12} + 171807 p^{3} T^{13} - 29328 p^{4} T^{14} + 11233 p^{5} T^{15} + 2470 p^{6} T^{16} - 239 p^{7} T^{17} + 4 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 6 T - 115 T^{2} - 1102 T^{3} + 4464 T^{4} + 68224 T^{5} - 80081 T^{6} - 1389198 T^{7} + 10527863 T^{8} + 1010702 T^{9} - 878583588 T^{10} + 1010702 p T^{11} + 10527863 p^{2} T^{12} - 1389198 p^{3} T^{13} - 80081 p^{4} T^{14} + 68224 p^{5} T^{15} + 4464 p^{6} T^{16} - 1102 p^{7} T^{17} - 115 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( ( 1 + 121 T^{2} + 132 T^{3} + 9910 T^{4} + 4344 T^{5} + 9910 p T^{6} + 132 p^{2} T^{7} + 121 p^{3} T^{8} + p^{5} T^{10} )^{2} \) | |
| 59 | \( 1 + 13 T + 8 T^{2} - 99 T^{3} - 1386 T^{4} - 71319 T^{5} - 553188 T^{6} - 1402305 T^{7} + 12050733 T^{8} + 238436626 T^{9} + 2061069616 T^{10} + 238436626 p T^{11} + 12050733 p^{2} T^{12} - 1402305 p^{3} T^{13} - 553188 p^{4} T^{14} - 71319 p^{5} T^{15} - 1386 p^{6} T^{16} - 99 p^{7} T^{17} + 8 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 10 T - 29 T^{2} - 586 T^{3} - 4096 T^{4} - 18694 T^{5} - 74095 T^{6} - 1489618 T^{7} - 1883845 T^{8} + 159588968 T^{9} + 1829886464 T^{10} + 159588968 p T^{11} - 1883845 p^{2} T^{12} - 1489618 p^{3} T^{13} - 74095 p^{4} T^{14} - 18694 p^{5} T^{15} - 4096 p^{6} T^{16} - 586 p^{7} T^{17} - 29 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 17 T - 86 T^{2} - 2045 T^{3} + 16832 T^{4} + 214933 T^{5} - 1861648 T^{6} - 10584233 T^{7} + 211377539 T^{8} + 361755136 T^{9} - 15322470844 T^{10} + 361755136 p T^{11} + 211377539 p^{2} T^{12} - 10584233 p^{3} T^{13} - 1861648 p^{4} T^{14} + 214933 p^{5} T^{15} + 16832 p^{6} T^{16} - 2045 p^{7} T^{17} - 86 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( ( 1 - 4 T + 191 T^{2} - 56 T^{3} + 17098 T^{4} + 17904 T^{5} + 17098 p T^{6} - 56 p^{2} T^{7} + 191 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 73 | \( ( 1 + 17 T + 441 T^{2} + 4824 T^{3} + 68238 T^{4} + 520146 T^{5} + 68238 p T^{6} + 4824 p^{2} T^{7} + 441 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 79 | \( 1 + 6 T - 71 T^{2} + 1534 T^{3} + 10236 T^{4} - 156164 T^{5} + 767899 T^{6} + 10545042 T^{7} - 131457605 T^{8} - 414891970 T^{9} + 8190118668 T^{10} - 414891970 p T^{11} - 131457605 p^{2} T^{12} + 10545042 p^{3} T^{13} + 767899 p^{4} T^{14} - 156164 p^{5} T^{15} + 10236 p^{6} T^{16} + 1534 p^{7} T^{17} - 71 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 12 T - 253 T^{2} + 2040 T^{3} + 53448 T^{4} - 232746 T^{5} - 7884123 T^{6} + 17386056 T^{7} + 891611235 T^{8} - 597479826 T^{9} - 81429796884 T^{10} - 597479826 p T^{11} + 891611235 p^{2} T^{12} + 17386056 p^{3} T^{13} - 7884123 p^{4} T^{14} - 232746 p^{5} T^{15} + 53448 p^{6} T^{16} + 2040 p^{7} T^{17} - 253 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( ( 1 - 22 T + 365 T^{2} - 4444 T^{3} + 50454 T^{4} - 469972 T^{5} + 50454 p T^{6} - 4444 p^{2} T^{7} + 365 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 97 | \( 1 - 27 T + 214 T^{2} + 115 T^{3} - 15534 T^{4} + 374911 T^{5} - 6022184 T^{6} + 37100679 T^{7} + 194591941 T^{8} - 4862046286 T^{9} + 49414074132 T^{10} - 4862046286 p T^{11} + 194591941 p^{2} T^{12} + 37100679 p^{3} T^{13} - 6022184 p^{4} T^{14} + 374911 p^{5} T^{15} - 15534 p^{6} T^{16} + 115 p^{7} T^{17} + 214 p^{8} T^{18} - 27 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.34363556815537126674721168329, −3.30403453192159362953913258011, −3.27719281046246611122386608367, −3.21861364727229155540008341380, −3.16164956680491977847925532454, −3.03797571030321722501508655416, −2.99507724242045059902487234785, −2.90916328301978180133269865419, −2.82028260873861646918278258743, −2.54997365300528617175514615245, −2.46049545582802936500279799415, −2.26801090116287275569149377307, −2.13717814725539256076359151823, −2.05718503242246555835068592255, −2.00950959225664011746727743311, −1.80433867458279767738782749682, −1.79583791238134322887931577220, −1.40851980632449211888189076048, −1.37794574017931649999971790709, −1.05635430754212010391320755716, −0.942901503537792649888675241023, −0.77759481909765734659507511143, −0.57986009316009859005233750486, −0.50992145680409116692848253304, −0.35103356654322131478356691539, 0.35103356654322131478356691539, 0.50992145680409116692848253304, 0.57986009316009859005233750486, 0.77759481909765734659507511143, 0.942901503537792649888675241023, 1.05635430754212010391320755716, 1.37794574017931649999971790709, 1.40851980632449211888189076048, 1.79583791238134322887931577220, 1.80433867458279767738782749682, 2.00950959225664011746727743311, 2.05718503242246555835068592255, 2.13717814725539256076359151823, 2.26801090116287275569149377307, 2.46049545582802936500279799415, 2.54997365300528617175514615245, 2.82028260873861646918278258743, 2.90916328301978180133269865419, 2.99507724242045059902487234785, 3.03797571030321722501508655416, 3.16164956680491977847925532454, 3.21861364727229155540008341380, 3.27719281046246611122386608367, 3.30403453192159362953913258011, 3.34363556815537126674721168329
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.