Dirichlet series
| L(s) = 1 | − 5·2-s − 10·3-s + 8·4-s + 50·6-s + 3·7-s + 2·8-s + 55·9-s + 7·11-s − 80·12-s − 15·14-s − 25·16-s − 9·17-s − 275·18-s + 22·19-s − 30·21-s − 35·22-s − 14·23-s − 20·24-s − 220·27-s + 24·28-s + 4·29-s + 22·31-s + 34·32-s − 70·33-s + 45·34-s + 440·36-s − 37-s + ⋯ |
| L(s) = 1 | − 3.53·2-s − 5.77·3-s + 4·4-s + 20.4·6-s + 1.13·7-s + 0.707·8-s + 55/3·9-s + 2.11·11-s − 23.0·12-s − 4.00·14-s − 6.25·16-s − 2.18·17-s − 64.8·18-s + 5.04·19-s − 6.54·21-s − 7.46·22-s − 2.91·23-s − 4.08·24-s − 42.3·27-s + 4.53·28-s + 0.742·29-s + 3.95·31-s + 6.01·32-s − 12.1·33-s + 7.71·34-s + 73.3·36-s − 0.164·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20} \cdot 59^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20} \cdot 59^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 5^{20} \cdot 59^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.03321\times 10^{15}\) |
| Root analytic conductor: | \(5.94422\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 5^{20} \cdot 59^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2345883379\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2345883379\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T )^{10} \) |
| 5 | \( 1 \) | |
| 59 | \( ( 1 + T )^{10} \) | |
| good | 2 | \( 1 + 5 T + 17 T^{2} + 43 T^{3} + 47 p T^{4} + 183 T^{5} + 83 p^{2} T^{6} + 557 T^{7} + 883 T^{8} + 329 p^{2} T^{9} + 1899 T^{10} + 329 p^{3} T^{11} + 883 p^{2} T^{12} + 557 p^{3} T^{13} + 83 p^{6} T^{14} + 183 p^{5} T^{15} + 47 p^{7} T^{16} + 43 p^{7} T^{17} + 17 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 - 3 T + 31 T^{2} - 92 T^{3} + 584 T^{4} - 1487 T^{5} + 7522 T^{6} - 2481 p T^{7} + 10477 p T^{8} - 152217 T^{9} + 82206 p T^{10} - 152217 p T^{11} + 10477 p^{3} T^{12} - 2481 p^{4} T^{13} + 7522 p^{4} T^{14} - 1487 p^{5} T^{15} + 584 p^{6} T^{16} - 92 p^{7} T^{17} + 31 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 7 T + 69 T^{2} - 340 T^{3} + 2076 T^{4} - 8699 T^{5} + 41762 T^{6} - 156611 T^{7} + 5239 p^{2} T^{8} - 2153829 T^{9} + 7698278 T^{10} - 2153829 p T^{11} + 5239 p^{4} T^{12} - 156611 p^{3} T^{13} + 41762 p^{4} T^{14} - 8699 p^{5} T^{15} + 2076 p^{6} T^{16} - 340 p^{7} T^{17} + 69 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 61 T^{2} - 56 T^{3} + 1778 T^{4} - 3758 T^{5} + 32030 T^{6} - 121390 T^{7} + 421997 T^{8} - 184772 p T^{9} + 5184794 T^{10} - 184772 p^{2} T^{11} + 421997 p^{2} T^{12} - 121390 p^{3} T^{13} + 32030 p^{4} T^{14} - 3758 p^{5} T^{15} + 1778 p^{6} T^{16} - 56 p^{7} T^{17} + 61 p^{8} T^{18} + p^{10} T^{20} \) | |
| 17 | \( 1 + 9 T + 103 T^{2} + 660 T^{3} + 4736 T^{4} + 25213 T^{5} + 144130 T^{6} + 674151 T^{7} + 3337407 T^{8} + 14138383 T^{9} + 62704798 T^{10} + 14138383 p T^{11} + 3337407 p^{2} T^{12} + 674151 p^{3} T^{13} + 144130 p^{4} T^{14} + 25213 p^{5} T^{15} + 4736 p^{6} T^{16} + 660 p^{7} T^{17} + 103 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 22 T + 350 T^{2} - 4040 T^{3} + 38920 T^{4} - 315738 T^{5} + 2248842 T^{6} - 14122034 T^{7} + 79635831 T^{8} - 403212370 T^{9} + 1850118512 T^{10} - 403212370 p T^{11} + 79635831 p^{2} T^{12} - 14122034 p^{3} T^{13} + 2248842 p^{4} T^{14} - 315738 p^{5} T^{15} + 38920 p^{6} T^{16} - 4040 p^{7} T^{17} + 350 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 14 T + 208 T^{2} + 1938 T^{3} + 17494 T^{4} + 123850 T^{5} + 850338 T^{6} + 4943958 T^{7} + 1233639 p T^{8} + 143469736 T^{9} + 727648456 T^{10} + 143469736 p T^{11} + 1233639 p^{3} T^{12} + 4943958 p^{3} T^{13} + 850338 p^{4} T^{14} + 123850 p^{5} T^{15} + 17494 p^{6} T^{16} + 1938 p^{7} T^{17} + 208 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 4 T + 220 T^{2} - 876 T^{3} + 802 p T^{4} - 88300 T^{5} + 1553606 T^{6} - 5415096 T^{7} + 72377049 T^{8} - 223896584 T^{9} + 2448983432 T^{10} - 223896584 p T^{11} + 72377049 p^{2} T^{12} - 5415096 p^{3} T^{13} + 1553606 p^{4} T^{14} - 88300 p^{5} T^{15} + 802 p^{7} T^{16} - 876 p^{7} T^{17} + 220 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 22 T + 399 T^{2} - 5181 T^{3} + 58775 T^{4} - 564096 T^{5} + 4866624 T^{6} - 37375051 T^{7} + 261660505 T^{8} - 1662803755 T^{9} + 9691553877 T^{10} - 1662803755 p T^{11} + 261660505 p^{2} T^{12} - 37375051 p^{3} T^{13} + 4866624 p^{4} T^{14} - 564096 p^{5} T^{15} + 58775 p^{6} T^{16} - 5181 p^{7} T^{17} + 399 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + T + 214 T^{2} + 331 T^{3} + 22442 T^{4} + 48936 T^{5} + 1552270 T^{6} + 4188940 T^{7} + 80121245 T^{8} + 230462068 T^{9} + 3288912796 T^{10} + 230462068 p T^{11} + 80121245 p^{2} T^{12} + 4188940 p^{3} T^{13} + 1552270 p^{4} T^{14} + 48936 p^{5} T^{15} + 22442 p^{6} T^{16} + 331 p^{7} T^{17} + 214 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 15 T + 374 T^{2} - 4329 T^{3} + 63118 T^{4} - 592454 T^{5} + 6409918 T^{6} - 50274474 T^{7} + 438308061 T^{8} - 2911759004 T^{9} + 21209065308 T^{10} - 2911759004 p T^{11} + 438308061 p^{2} T^{12} - 50274474 p^{3} T^{13} + 6409918 p^{4} T^{14} - 592454 p^{5} T^{15} + 63118 p^{6} T^{16} - 4329 p^{7} T^{17} + 374 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 6 T + 332 T^{2} - 1858 T^{3} + 52638 T^{4} - 270110 T^{5} + 5246841 T^{6} - 24212190 T^{7} + 363451615 T^{8} - 1475746746 T^{9} + 18248073017 T^{10} - 1475746746 p T^{11} + 363451615 p^{2} T^{12} - 24212190 p^{3} T^{13} + 5246841 p^{4} T^{14} - 270110 p^{5} T^{15} + 52638 p^{6} T^{16} - 1858 p^{7} T^{17} + 332 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 11 T + 296 T^{2} + 3009 T^{3} + 43754 T^{4} + 402530 T^{5} + 4236562 T^{6} + 35057214 T^{7} + 297706645 T^{8} + 2202429618 T^{9} + 15921234908 T^{10} + 2202429618 p T^{11} + 297706645 p^{2} T^{12} + 35057214 p^{3} T^{13} + 4236562 p^{4} T^{14} + 402530 p^{5} T^{15} + 43754 p^{6} T^{16} + 3009 p^{7} T^{17} + 296 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 7 T + 340 T^{2} + 2831 T^{3} + 57950 T^{4} + 504184 T^{5} + 6632626 T^{6} + 53772776 T^{7} + 554108753 T^{8} + 73985742 p T^{9} + 34199304452 T^{10} + 73985742 p^{2} T^{11} + 554108753 p^{2} T^{12} + 53772776 p^{3} T^{13} + 6632626 p^{4} T^{14} + 504184 p^{5} T^{15} + 57950 p^{6} T^{16} + 2831 p^{7} T^{17} + 340 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 14 T + 364 T^{2} - 4541 T^{3} + 64176 T^{4} - 704759 T^{5} + 7471918 T^{6} - 71248769 T^{7} + 651562863 T^{8} - 5405501713 T^{9} + 44619000368 T^{10} - 5405501713 p T^{11} + 651562863 p^{2} T^{12} - 71248769 p^{3} T^{13} + 7471918 p^{4} T^{14} - 704759 p^{5} T^{15} + 64176 p^{6} T^{16} - 4541 p^{7} T^{17} + 364 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 11 T + 488 T^{2} - 5107 T^{3} + 115400 T^{4} - 1127918 T^{5} + 17396434 T^{6} - 154892366 T^{7} + 1845463519 T^{8} - 14583206708 T^{9} + 143564298888 T^{10} - 14583206708 p T^{11} + 1845463519 p^{2} T^{12} - 154892366 p^{3} T^{13} + 17396434 p^{4} T^{14} - 1127918 p^{5} T^{15} + 115400 p^{6} T^{16} - 5107 p^{7} T^{17} + 488 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 27 T + 811 T^{2} - 14132 T^{3} + 250715 T^{4} - 3307766 T^{5} + 43829056 T^{6} - 471262289 T^{7} + 71638159 p T^{8} - 46225433410 T^{9} + 422206939359 T^{10} - 46225433410 p T^{11} + 71638159 p^{3} T^{12} - 471262289 p^{3} T^{13} + 43829056 p^{4} T^{14} - 3307766 p^{5} T^{15} + 250715 p^{6} T^{16} - 14132 p^{7} T^{17} + 811 p^{8} T^{18} - 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 22 T + 476 T^{2} - 6227 T^{3} + 91692 T^{4} - 1031119 T^{5} + 12825466 T^{6} - 125777197 T^{7} + 1336129507 T^{8} - 11584033083 T^{9} + 109182064628 T^{10} - 11584033083 p T^{11} + 1336129507 p^{2} T^{12} - 125777197 p^{3} T^{13} + 12825466 p^{4} T^{14} - 1031119 p^{5} T^{15} + 91692 p^{6} T^{16} - 6227 p^{7} T^{17} + 476 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 11 T + 577 T^{2} - 5632 T^{3} + 159562 T^{4} - 1376975 T^{5} + 27979458 T^{6} - 212878875 T^{7} + 3457016869 T^{8} - 23067795217 T^{9} + 315514254506 T^{10} - 23067795217 p T^{11} + 3457016869 p^{2} T^{12} - 212878875 p^{3} T^{13} + 27979458 p^{4} T^{14} - 1376975 p^{5} T^{15} + 159562 p^{6} T^{16} - 5632 p^{7} T^{17} + 577 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 9 T + 473 T^{2} + 3654 T^{3} + 92394 T^{4} + 612543 T^{5} + 9209770 T^{6} + 53905291 T^{7} + 470954713 T^{8} + 3113766761 T^{9} + 19923312366 T^{10} + 3113766761 p T^{11} + 470954713 p^{2} T^{12} + 53905291 p^{3} T^{13} + 9209770 p^{4} T^{14} + 612543 p^{5} T^{15} + 92394 p^{6} T^{16} + 3654 p^{7} T^{17} + 473 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 7 T + 633 T^{2} - 5420 T^{3} + 194735 T^{4} - 1791208 T^{5} + 38833838 T^{6} - 345394233 T^{7} + 5511578651 T^{8} - 44073791648 T^{9} + 572156738379 T^{10} - 44073791648 p T^{11} + 5511578651 p^{2} T^{12} - 345394233 p^{3} T^{13} + 38833838 p^{4} T^{14} - 1791208 p^{5} T^{15} + 194735 p^{6} T^{16} - 5420 p^{7} T^{17} + 633 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 8 T + 510 T^{2} + 4653 T^{3} + 146616 T^{4} + 1289905 T^{5} + 28879950 T^{6} + 234936775 T^{7} + 4186236231 T^{8} + 30702012415 T^{9} + 462936569236 T^{10} + 30702012415 p T^{11} + 4186236231 p^{2} T^{12} + 234936775 p^{3} T^{13} + 28879950 p^{4} T^{14} + 1289905 p^{5} T^{15} + 146616 p^{6} T^{16} + 4653 p^{7} T^{17} + 510 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−2.72381363920950055992334604559, −2.59520577765135922126289757920, −2.53529041954280744638965753839, −2.47435738393194906452599344735, −2.39705154952316394478497213021, −2.34143468311228227340043316045, −2.08148201397528654875490409258, −1.98745327806757954307448814942, −1.76491210299504038965810925535, −1.73268413832418151505193369099, −1.67371968580487360618765270920, −1.61882800866723556282534576979, −1.51003340183264283685335348338, −1.34901271272078461320374627779, −1.29807794560390446203853266370, −1.18334596835940852224350162018, −0.943554540205346051404001146659, −0.919501673661553557880787424978, −0.837302052766415783633892373188, −0.58558052837451011264236999434, −0.55751479338444453714430973703, −0.54976198164097123366443881836, −0.51075270922621825837368866143, −0.38213220372729192878623146518, −0.19806548353337351077918152011, 0.19806548353337351077918152011, 0.38213220372729192878623146518, 0.51075270922621825837368866143, 0.54976198164097123366443881836, 0.55751479338444453714430973703, 0.58558052837451011264236999434, 0.837302052766415783633892373188, 0.919501673661553557880787424978, 0.943554540205346051404001146659, 1.18334596835940852224350162018, 1.29807794560390446203853266370, 1.34901271272078461320374627779, 1.51003340183264283685335348338, 1.61882800866723556282534576979, 1.67371968580487360618765270920, 1.73268413832418151505193369099, 1.76491210299504038965810925535, 1.98745327806757954307448814942, 2.08148201397528654875490409258, 2.34143468311228227340043316045, 2.39705154952316394478497213021, 2.47435738393194906452599344735, 2.53529041954280744638965753839, 2.59520577765135922126289757920, 2.72381363920950055992334604559
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.