Dirichlet series
| L(s) = 1 | − 3·4-s + 3·7-s + 6·13-s + 5·16-s − 12·17-s + 3·19-s − 32·25-s + 9·27-s − 9·28-s − 15·29-s − 9·31-s + 6·32-s + 6·37-s − 9·41-s + 3·43-s + 15·47-s − 7·49-s − 18·52-s − 9·53-s − 18·59-s + 12·61-s − 64-s − 10·67-s + 36·68-s + 3·73-s − 9·76-s + 20·79-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 1.13·7-s + 1.66·13-s + 5/4·16-s − 2.91·17-s + 0.688·19-s − 6.39·25-s + 1.73·27-s − 1.70·28-s − 2.78·29-s − 1.61·31-s + 1.06·32-s + 0.986·37-s − 1.40·41-s + 0.457·43-s + 2.18·47-s − 49-s − 2.49·52-s − 1.23·53-s − 2.34·59-s + 1.53·61-s − 1/8·64-s − 1.22·67-s + 4.36·68-s + 0.351·73-s − 1.03·76-s + 2.25·79-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{20} \cdot 7^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00103795\) |
| Root analytic conductor: | \(0.709265\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{20} \cdot 7^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2404682648\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2404682648\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - p^{2} T^{3} + 4 p T^{4} + p^{2} T^{5} + 4 p^{2} T^{6} - p^{4} T^{7} + p^{5} T^{10} \) |
| 7 | \( 1 - 3 T + 16 T^{2} - 62 T^{3} + 220 T^{4} - 473 T^{5} + 220 p T^{6} - 62 p^{2} T^{7} + 16 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 2 | \( 1 + 3 T^{2} + p^{2} T^{4} - 3 p T^{5} - p T^{6} + 3 T^{7} - 7 p T^{8} + 39 T^{9} + T^{10} + 39 p T^{11} - 7 p^{3} T^{12} + 3 p^{3} T^{13} - p^{5} T^{14} - 3 p^{6} T^{15} + p^{8} T^{16} + 3 p^{8} T^{18} + p^{10} T^{20} \) |
| 5 | \( ( 1 + 16 T^{2} - 6 T^{3} + 127 T^{4} - 51 T^{5} + 127 p T^{6} - 6 p^{2} T^{7} + 16 p^{3} T^{8} + p^{5} T^{10} )^{2} \) | |
| 11 | \( 1 - 6 p T^{2} + 2257 T^{4} - 51461 T^{6} + 77732 p T^{8} - 10752323 T^{10} + 77732 p^{3} T^{12} - 51461 p^{4} T^{14} + 2257 p^{6} T^{16} - 6 p^{9} T^{18} + p^{10} T^{20} \) | |
| 13 | \( 1 - 6 T + 68 T^{2} - 336 T^{3} + 2292 T^{4} - 723 p T^{5} + 51837 T^{6} - 187401 T^{7} + 909867 T^{8} - 3004662 T^{9} + 13054461 T^{10} - 3004662 p T^{11} + 909867 p^{2} T^{12} - 187401 p^{3} T^{13} + 51837 p^{4} T^{14} - 723 p^{6} T^{15} + 2292 p^{6} T^{16} - 336 p^{7} T^{17} + 68 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 12 T + 26 T^{2} - 36 T^{3} + 1143 T^{4} + 5247 T^{5} - 21540 T^{6} - 73476 T^{7} + 337539 T^{8} - 599625 T^{9} - 13374333 T^{10} - 599625 p T^{11} + 337539 p^{2} T^{12} - 73476 p^{3} T^{13} - 21540 p^{4} T^{14} + 5247 p^{5} T^{15} + 1143 p^{6} T^{16} - 36 p^{7} T^{17} + 26 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 3 T + 71 T^{2} - 204 T^{3} + 2646 T^{4} - 8547 T^{5} + 74607 T^{6} - 258954 T^{7} + 1743726 T^{8} - 5989488 T^{9} + 35261703 T^{10} - 5989488 p T^{11} + 1743726 p^{2} T^{12} - 258954 p^{3} T^{13} + 74607 p^{4} T^{14} - 8547 p^{5} T^{15} + 2646 p^{6} T^{16} - 204 p^{7} T^{17} + 71 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 165 T^{2} + 13144 T^{4} - 668429 T^{6} + 24050461 T^{8} - 639494549 T^{10} + 24050461 p^{2} T^{12} - 668429 p^{4} T^{14} + 13144 p^{6} T^{16} - 165 p^{8} T^{18} + p^{10} T^{20} \) | |
| 29 | \( 1 + 15 T + 150 T^{2} + 1125 T^{3} + 6691 T^{4} + 30108 T^{5} + 81631 T^{6} - 232971 T^{7} - 5137202 T^{8} - 44535417 T^{9} - 275752187 T^{10} - 44535417 p T^{11} - 5137202 p^{2} T^{12} - 232971 p^{3} T^{13} + 81631 p^{4} T^{14} + 30108 p^{5} T^{15} + 6691 p^{6} T^{16} + 1125 p^{7} T^{17} + 150 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 9 T + 149 T^{2} + 1098 T^{3} + 10878 T^{4} + 60723 T^{5} + 461409 T^{6} + 2027286 T^{7} + 13421802 T^{8} + 50078664 T^{9} + 374531595 T^{10} + 50078664 p T^{11} + 13421802 p^{2} T^{12} + 2027286 p^{3} T^{13} + 461409 p^{4} T^{14} + 60723 p^{5} T^{15} + 10878 p^{6} T^{16} + 1098 p^{7} T^{17} + 149 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 6 T - 97 T^{2} + 194 T^{3} + 7179 T^{4} + 3556 T^{5} - 323794 T^{6} - 533292 T^{7} + 10739317 T^{8} + 10946526 T^{9} - 345629139 T^{10} + 10946526 p T^{11} + 10739317 p^{2} T^{12} - 533292 p^{3} T^{13} - 323794 p^{4} T^{14} + 3556 p^{5} T^{15} + 7179 p^{6} T^{16} + 194 p^{7} T^{17} - 97 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 9 T - 34 T^{2} - 747 T^{3} - 2085 T^{4} + 20394 T^{5} + 110775 T^{6} - 15219 p T^{7} - 5992218 T^{8} + 18494757 T^{9} + 381591615 T^{10} + 18494757 p T^{11} - 5992218 p^{2} T^{12} - 15219 p^{4} T^{13} + 110775 p^{4} T^{14} + 20394 p^{5} T^{15} - 2085 p^{6} T^{16} - 747 p^{7} T^{17} - 34 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 3 T - 79 T^{2} + 1100 T^{3} + 1674 T^{4} - 79931 T^{5} + 324899 T^{6} + 75114 p T^{7} - 28512986 T^{8} - 52724394 T^{9} + 1438527201 T^{10} - 52724394 p T^{11} - 28512986 p^{2} T^{12} + 75114 p^{4} T^{13} + 324899 p^{4} T^{14} - 79931 p^{5} T^{15} + 1674 p^{6} T^{16} + 1100 p^{7} T^{17} - 79 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 15 T - 49 T^{2} + 1068 T^{3} + 9486 T^{4} - 83265 T^{5} - 760914 T^{6} + 61443 p T^{7} + 57371082 T^{8} - 59131839 T^{9} - 3026317959 T^{10} - 59131839 p T^{11} + 57371082 p^{2} T^{12} + 61443 p^{4} T^{13} - 760914 p^{4} T^{14} - 83265 p^{5} T^{15} + 9486 p^{6} T^{16} + 1068 p^{7} T^{17} - 49 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 9 T + 237 T^{2} + 1890 T^{3} + 28720 T^{4} + 208689 T^{5} + 2523571 T^{6} + 16852668 T^{7} + 178203742 T^{8} + 1088604978 T^{9} + 10361882797 T^{10} + 1088604978 p T^{11} + 178203742 p^{2} T^{12} + 16852668 p^{3} T^{13} + 2523571 p^{4} T^{14} + 208689 p^{5} T^{15} + 28720 p^{6} T^{16} + 1890 p^{7} T^{17} + 237 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 18 T - 55 T^{2} - 1536 T^{3} + 19971 T^{4} + 205494 T^{5} - 1764945 T^{6} - 8798931 T^{7} + 181121100 T^{8} + 308804295 T^{9} - 11121159681 T^{10} + 308804295 p T^{11} + 181121100 p^{2} T^{12} - 8798931 p^{3} T^{13} - 1764945 p^{4} T^{14} + 205494 p^{5} T^{15} + 19971 p^{6} T^{16} - 1536 p^{7} T^{17} - 55 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 12 T + 251 T^{2} - 2436 T^{3} + 34779 T^{4} - 347730 T^{5} + 3716049 T^{6} - 34819755 T^{7} + 301038894 T^{8} - 2709237273 T^{9} + 20488848807 T^{10} - 2709237273 p T^{11} + 301038894 p^{2} T^{12} - 34819755 p^{3} T^{13} + 3716049 p^{4} T^{14} - 347730 p^{5} T^{15} + 34779 p^{6} T^{16} - 2436 p^{7} T^{17} + 251 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 10 T - 182 T^{2} - 2448 T^{3} + 18867 T^{4} + 319605 T^{5} - 772530 T^{6} - 23213154 T^{7} - 12219093 T^{8} + 697872289 T^{9} + 3674653819 T^{10} + 697872289 p T^{11} - 12219093 p^{2} T^{12} - 23213154 p^{3} T^{13} - 772530 p^{4} T^{14} + 319605 p^{5} T^{15} + 18867 p^{6} T^{16} - 2448 p^{7} T^{17} - 182 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 351 T^{2} + 63037 T^{4} - 7800935 T^{6} + 747809113 T^{8} - 58386380555 T^{10} + 747809113 p^{2} T^{12} - 7800935 p^{4} T^{14} + 63037 p^{6} T^{16} - 351 p^{8} T^{18} + p^{10} T^{20} \) | |
| 73 | \( 1 - 3 T + 260 T^{2} - 771 T^{3} + 36639 T^{4} - 155724 T^{5} + 3663555 T^{6} - 21043473 T^{7} + 293486934 T^{8} - 2074196103 T^{9} + 21799556757 T^{10} - 2074196103 p T^{11} + 293486934 p^{2} T^{12} - 21043473 p^{3} T^{13} + 3663555 p^{4} T^{14} - 155724 p^{5} T^{15} + 36639 p^{6} T^{16} - 771 p^{7} T^{17} + 260 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 20 T + 46 T^{2} - 144 T^{3} + 21153 T^{4} - 101181 T^{5} - 106944 T^{6} - 9264000 T^{7} + 7962453 T^{8} + 230795113 T^{9} + 4723714795 T^{10} + 230795113 p T^{11} + 7962453 p^{2} T^{12} - 9264000 p^{3} T^{13} - 106944 p^{4} T^{14} - 101181 p^{5} T^{15} + 21153 p^{6} T^{16} - 144 p^{7} T^{17} + 46 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 15 T - 136 T^{2} - 1773 T^{3} + 22674 T^{4} + 93717 T^{5} - 3687774 T^{6} - 10067337 T^{7} + 346135869 T^{8} + 496605294 T^{9} - 27460905396 T^{10} + 496605294 p T^{11} + 346135869 p^{2} T^{12} - 10067337 p^{3} T^{13} - 3687774 p^{4} T^{14} + 93717 p^{5} T^{15} + 22674 p^{6} T^{16} - 1773 p^{7} T^{17} - 136 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 24 T + 44 T^{2} + 2592 T^{3} - 1287 T^{4} - 278721 T^{5} - 235110 T^{6} + 13705920 T^{7} + 186157425 T^{8} - 904992183 T^{9} - 14602879521 T^{10} - 904992183 p T^{11} + 186157425 p^{2} T^{12} + 13705920 p^{3} T^{13} - 235110 p^{4} T^{14} - 278721 p^{5} T^{15} - 1287 p^{6} T^{16} + 2592 p^{7} T^{17} + 44 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 6 T + 311 T^{2} - 1794 T^{3} + 51903 T^{4} - 385032 T^{5} + 6010353 T^{6} - 63309837 T^{7} + 574248354 T^{8} - 8264282925 T^{9} + 54719955099 T^{10} - 8264282925 p T^{11} + 574248354 p^{2} T^{12} - 63309837 p^{3} T^{13} + 6010353 p^{4} T^{14} - 385032 p^{5} T^{15} + 51903 p^{6} T^{16} - 1794 p^{7} T^{17} + 311 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−6.27955897011664389080273747741, −6.27578699173217066747188596408, −6.10393165918656309466994673722, −5.84209608895373911285051621671, −5.81241980725022232798651687315, −5.69232466547016936706698071670, −5.44622812072426785487193772882, −5.30410482081078681096183615632, −4.96097361162037598788362102451, −4.81569658156642125329987650892, −4.66078822304143354051068806856, −4.65949997762408433700416045833, −4.39620690277133779342359245136, −4.25949135515091530670074020028, −4.12982533312461995788384839088, −3.80504370136066508778738496390, −3.56680784594110894288333789375, −3.47025791933994983015690697732, −3.42728543789873947427918259141, −3.39028504049013401082288020391, −2.53390167019242355721496667255, −2.19953178463900688028247068286, −2.08008475143022196357803850682, −1.94896097473872960324454837157, −1.58891858178962163003513947640, 1.58891858178962163003513947640, 1.94896097473872960324454837157, 2.08008475143022196357803850682, 2.19953178463900688028247068286, 2.53390167019242355721496667255, 3.39028504049013401082288020391, 3.42728543789873947427918259141, 3.47025791933994983015690697732, 3.56680784594110894288333789375, 3.80504370136066508778738496390, 4.12982533312461995788384839088, 4.25949135515091530670074020028, 4.39620690277133779342359245136, 4.65949997762408433700416045833, 4.66078822304143354051068806856, 4.81569658156642125329987650892, 4.96097361162037598788362102451, 5.30410482081078681096183615632, 5.44622812072426785487193772882, 5.69232466547016936706698071670, 5.81241980725022232798651687315, 5.84209608895373911285051621671, 6.10393165918656309466994673722, 6.27578699173217066747188596408, 6.27955897011664389080273747741
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.