Dirichlet series
| L(s) = 1 | − 3·4-s − 12·11-s − 6·13-s + 5·16-s + 24·17-s − 15·23-s + 16·25-s + 15·29-s − 9·31-s + 6·32-s − 12·37-s − 9·41-s + 3·43-s + 36·44-s + 15·47-s + 18·52-s − 18·59-s + 12·61-s − 64-s − 10·67-s − 72·68-s + 20·79-s − 15·83-s − 48·89-s + 45·92-s − 6·97-s − 48·100-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 3.61·11-s − 1.66·13-s + 5/4·16-s + 5.82·17-s − 3.12·23-s + 16/5·25-s + 2.78·29-s − 1.61·31-s + 1.06·32-s − 1.97·37-s − 1.40·41-s + 0.457·43-s + 5.42·44-s + 2.18·47-s + 2.49·52-s − 2.34·59-s + 1.53·61-s − 1/8·64-s − 1.22·67-s − 8.73·68-s + 2.25·79-s − 1.64·83-s − 5.08·89-s + 4.69·92-s − 0.609·97-s − 4.79·100-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{30} \cdot 7^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.73128\times 10^{10}\) |
| Root analytic conductor: | \(3.25026\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{30} \cdot 7^{20} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.01834950050\) |
| \(L(\frac12)\) | \(\approx\) | \(0.01834950050\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) | |
| 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 | \( 1 - 16 T^{2} - 12 T^{3} + 129 T^{4} + 147 T^{5} - 726 T^{6} - 144 p T^{7} + 3663 T^{8} + 1257 T^{9} - 18069 T^{10} + 1257 p T^{11} + 3663 p^{2} T^{12} - 144 p^{4} T^{13} - 726 p^{4} T^{14} + 147 p^{5} T^{15} + 129 p^{6} T^{16} - 12 p^{7} T^{17} - 16 p^{8} T^{18} + p^{10} T^{20} \) | |
| 11 | \( 1 + 12 T + 105 T^{2} + 684 T^{3} + 3745 T^{4} + 17382 T^{5} + 72307 T^{6} + 270561 T^{7} + 952558 T^{8} + 3194589 T^{9} + 10673917 T^{10} + 3194589 p T^{11} + 952558 p^{2} T^{12} + 270561 p^{3} T^{13} + 72307 p^{4} T^{14} + 17382 p^{5} T^{15} + 3745 p^{6} T^{16} + 684 p^{7} T^{17} + 105 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 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 | \( ( 1 - 12 T + 118 T^{2} - 726 T^{3} + 4069 T^{4} - 17235 T^{5} + 4069 p T^{6} - 726 p^{2} T^{7} + 118 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 19 | \( 1 - 7 p T^{2} + 8580 T^{4} - 354546 T^{6} + 10419720 T^{8} - 227952213 T^{10} + 10419720 p^{2} T^{12} - 354546 p^{4} T^{14} + 8580 p^{6} T^{16} - 7 p^{9} T^{18} + p^{10} T^{20} \) | |
| 23 | \( 1 + 15 T + 195 T^{2} + 1800 T^{3} + 15028 T^{4} + 107121 T^{5} + 712306 T^{6} + 4264599 T^{7} + 24178468 T^{8} + 125954103 T^{9} + 628044835 T^{10} + 125954103 p T^{11} + 24178468 p^{2} T^{12} + 4264599 p^{3} T^{13} + 712306 p^{4} T^{14} + 107121 p^{5} T^{15} + 15028 p^{6} T^{16} + 1800 p^{7} T^{17} + 195 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 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 | \( ( 1 + 6 T + 133 T^{2} + 496 T^{3} + 7534 T^{4} + 20884 T^{5} + 7534 p T^{6} + 496 p^{2} T^{7} + 133 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 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 | \( 1 - 393 T^{2} + 74860 T^{4} - 9044030 T^{6} + 764489716 T^{8} - 47150015153 T^{10} + 764489716 p^{2} T^{12} - 9044030 p^{4} T^{14} + 74860 p^{6} T^{16} - 393 p^{8} T^{18} + p^{10} T^{20} \) | |
| 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 | \( 1 - 7 p T^{2} + 124869 T^{4} - 19478727 T^{6} + 2170586121 T^{8} - 181575980223 T^{10} + 2170586121 p^{2} T^{12} - 19478727 p^{4} T^{14} + 124869 p^{6} T^{16} - 7 p^{9} T^{18} + p^{10} T^{20} \) | |
| 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 | \( ( 1 + 24 T + 532 T^{2} + 7680 T^{3} + 99991 T^{4} + 992529 T^{5} + 99991 p T^{6} + 7680 p^{2} T^{7} + 532 p^{3} T^{8} + 24 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 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
−3.54913152693074962151419867222, −3.26025164085007166841613895487, −3.13808779889020975340504026859, −3.11045496472035502636241332699, −3.06429153059450682307614014419, −2.87898240359564145891895896251, −2.73823894941580358654853861647, −2.71195474888217828530905690864, −2.71168509696724299372864508610, −2.56395206943660512842373511302, −2.49928225857984293751525560521, −2.37338275827050266595700232326, −2.32589571629954348815105822733, −1.82668429366951695290917973168, −1.74828047137906055420398678138, −1.64240301211239038508646758404, −1.58022432492689337069528157752, −1.48830590110788396782177042474, −1.07804430829019615597331442439, −1.07035084401307856404532069257, −1.03731346991581652496362093993, −0.876268404524141113093470062423, −0.55198153948065139759623053917, −0.12527237557891822934794697130, −0.04500262606857755484307233916, 0.04500262606857755484307233916, 0.12527237557891822934794697130, 0.55198153948065139759623053917, 0.876268404524141113093470062423, 1.03731346991581652496362093993, 1.07035084401307856404532069257, 1.07804430829019615597331442439, 1.48830590110788396782177042474, 1.58022432492689337069528157752, 1.64240301211239038508646758404, 1.74828047137906055420398678138, 1.82668429366951695290917973168, 2.32589571629954348815105822733, 2.37338275827050266595700232326, 2.49928225857984293751525560521, 2.56395206943660512842373511302, 2.71168509696724299372864508610, 2.71195474888217828530905690864, 2.73823894941580358654853861647, 2.87898240359564145891895896251, 3.06429153059450682307614014419, 3.11045496472035502636241332699, 3.13808779889020975340504026859, 3.26025164085007166841613895487, 3.54913152693074962151419867222
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.