Dirichlet series
| L(s) = 1 | − 3-s + 5·7-s + 2·9-s + 6·11-s + 6·17-s + 3·19-s − 5·21-s + 24·23-s + 27-s + 15·31-s − 6·33-s + 37-s + 8·41-s + 26·43-s + 14·47-s + 6·49-s − 6·51-s − 24·53-s − 3·57-s + 42·61-s + 10·63-s − 7·67-s − 24·69-s + 3·73-s + 30·77-s + 79-s − 4·81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.88·7-s + 2/3·9-s + 1.80·11-s + 1.45·17-s + 0.688·19-s − 1.09·21-s + 5.00·23-s + 0.192·27-s + 2.69·31-s − 1.04·33-s + 0.164·37-s + 1.24·41-s + 3.96·43-s + 2.04·47-s + 6/7·49-s − 0.840·51-s − 3.29·53-s − 0.397·57-s + 5.37·61-s + 1.25·63-s − 0.855·67-s − 2.88·69-s + 0.351·73-s + 3.41·77-s + 0.112·79-s − 4/9·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 5^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 5^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{20} \cdot 3^{10} \cdot 5^{20} \cdot 7^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.75777\times 10^{12}\) |
| Root analytic conductor: | \(4.09494\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{20} \cdot 3^{10} \cdot 5^{20} \cdot 7^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(88.69324647\) |
| \(L(\frac12)\) | \(\approx\) | \(88.69324647\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + T - T^{2} - 4 T^{3} + T^{4} + 7 p T^{5} + p T^{6} - 4 p^{2} T^{7} - p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - 5 T + 19 T^{2} - 6 T^{3} - 13 p T^{4} + 533 T^{5} - 13 p^{2} T^{6} - 6 p^{2} T^{7} + 19 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 11 | \( 1 - 6 T + 51 T^{2} - 234 T^{3} + 1195 T^{4} - 3666 T^{5} + 12622 T^{6} - 19434 T^{7} + 25645 T^{8} + 184704 T^{9} - 509783 T^{10} + 184704 p T^{11} + 25645 p^{2} T^{12} - 19434 p^{3} T^{13} + 12622 p^{4} T^{14} - 3666 p^{5} T^{15} + 1195 p^{6} T^{16} - 234 p^{7} T^{17} + 51 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) |
| 13 | \( 1 - 49 T^{2} + 1364 T^{4} - 28371 T^{6} + 474775 T^{8} - 6687888 T^{10} + 474775 p^{2} T^{12} - 28371 p^{4} T^{14} + 1364 p^{6} T^{16} - 49 p^{8} T^{18} + p^{10} T^{20} \) | |
| 17 | \( 1 - 6 T - 15 T^{2} + 138 T^{3} + 193 T^{4} - 162 p T^{5} + 4928 T^{6} + 30306 T^{7} - 220151 T^{8} - 325680 T^{9} + 6075881 T^{10} - 325680 p T^{11} - 220151 p^{2} T^{12} + 30306 p^{3} T^{13} + 4928 p^{4} T^{14} - 162 p^{6} T^{15} + 193 p^{6} T^{16} + 138 p^{7} T^{17} - 15 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 3 T + 83 T^{2} - 240 T^{3} + 200 p T^{4} - 10380 T^{5} + 124845 T^{6} - 315315 T^{7} + 3213415 T^{8} - 7312140 T^{9} + 67225536 T^{10} - 7312140 p T^{11} + 3213415 p^{2} T^{12} - 315315 p^{3} T^{13} + 124845 p^{4} T^{14} - 10380 p^{5} T^{15} + 200 p^{7} T^{16} - 240 p^{7} T^{17} + 83 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 24 T + 338 T^{2} - 3504 T^{3} + 29201 T^{4} - 203310 T^{5} + 1216412 T^{6} - 6430392 T^{7} + 31045441 T^{8} - 143820006 T^{9} + 675418266 T^{10} - 143820006 p T^{11} + 31045441 p^{2} T^{12} - 6430392 p^{3} T^{13} + 1216412 p^{4} T^{14} - 203310 p^{5} T^{15} + 29201 p^{6} T^{16} - 3504 p^{7} T^{17} + 338 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 148 T^{2} + 11438 T^{4} - 619594 T^{6} + 25499257 T^{8} - 826380936 T^{10} + 25499257 p^{2} T^{12} - 619594 p^{4} T^{14} + 11438 p^{6} T^{16} - 148 p^{8} T^{18} + p^{10} T^{20} \) | |
| 31 | \( 1 - 15 T + 167 T^{2} - 1380 T^{3} + 9668 T^{4} - 61404 T^{5} + 363093 T^{6} - 2129475 T^{7} + 12161347 T^{8} - 68552376 T^{9} + 387742776 T^{10} - 68552376 p T^{11} + 12161347 p^{2} T^{12} - 2129475 p^{3} T^{13} + 363093 p^{4} T^{14} - 61404 p^{5} T^{15} + 9668 p^{6} T^{16} - 1380 p^{7} T^{17} + 167 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - T - 117 T^{2} + 176 T^{3} + 6668 T^{4} - 11380 T^{5} - 299439 T^{6} + 438173 T^{7} + 12712375 T^{8} - 7307172 T^{9} - 492022188 T^{10} - 7307172 p T^{11} + 12712375 p^{2} T^{12} + 438173 p^{3} T^{13} - 299439 p^{4} T^{14} - 11380 p^{5} T^{15} + 6668 p^{6} T^{16} + 176 p^{7} T^{17} - 117 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( ( 1 - 4 T + 90 T^{2} - 592 T^{3} + 3749 T^{4} - 36434 T^{5} + 3749 p T^{6} - 592 p^{2} T^{7} + 90 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 43 | \( ( 1 - 13 T + 173 T^{2} - 1668 T^{3} + 14545 T^{4} - 96933 T^{5} + 14545 p T^{6} - 1668 p^{2} T^{7} + 173 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 47 | \( 1 - 14 T + 49 T^{2} + 70 T^{3} - 1449 T^{4} + 16780 T^{5} - 75362 T^{6} - 133028 T^{7} + 1885261 T^{8} - 57961722 T^{9} + 732803503 T^{10} - 57961722 p T^{11} + 1885261 p^{2} T^{12} - 133028 p^{3} T^{13} - 75362 p^{4} T^{14} + 16780 p^{5} T^{15} - 1449 p^{6} T^{16} + 70 p^{7} T^{17} + 49 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 24 T + 337 T^{2} + 3480 T^{3} + 28655 T^{4} + 224976 T^{5} + 1724958 T^{6} + 14318448 T^{7} + 120990445 T^{8} + 964554888 T^{9} + 7393360287 T^{10} + 964554888 p T^{11} + 120990445 p^{2} T^{12} + 14318448 p^{3} T^{13} + 1724958 p^{4} T^{14} + 224976 p^{5} T^{15} + 28655 p^{6} T^{16} + 3480 p^{7} T^{17} + 337 p^{8} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 135 T^{2} + 756 T^{3} + 7351 T^{4} - 84450 T^{5} - 41974 T^{6} + 3597210 T^{7} - 8726459 T^{8} - 56020614 T^{9} + 414268763 T^{10} - 56020614 p T^{11} - 8726459 p^{2} T^{12} + 3597210 p^{3} T^{13} - 41974 p^{4} T^{14} - 84450 p^{5} T^{15} + 7351 p^{6} T^{16} + 756 p^{7} T^{17} - 135 p^{8} T^{18} + p^{10} T^{20} \) | |
| 61 | \( 1 - 42 T + 1112 T^{2} - 22008 T^{3} + 361073 T^{4} - 5062200 T^{5} + 62481246 T^{6} - 687242394 T^{7} + 6821292673 T^{8} - 61325603712 T^{9} + 502052393310 T^{10} - 61325603712 p T^{11} + 6821292673 p^{2} T^{12} - 687242394 p^{3} T^{13} + 62481246 p^{4} T^{14} - 5062200 p^{5} T^{15} + 361073 p^{6} T^{16} - 22008 p^{7} T^{17} + 1112 p^{8} T^{18} - 42 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 7 T - 94 T^{2} - 1731 T^{3} + 26 T^{4} + 170241 T^{5} + 879508 T^{6} - 10247793 T^{7} - 116682655 T^{8} + 286659444 T^{9} + 9801469996 T^{10} + 286659444 p T^{11} - 116682655 p^{2} T^{12} - 10247793 p^{3} T^{13} + 879508 p^{4} T^{14} + 170241 p^{5} T^{15} + 26 p^{6} T^{16} - 1731 p^{7} T^{17} - 94 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 422 T^{2} + 92093 T^{4} - 13410012 T^{6} + 1429352290 T^{8} - 115763220252 T^{10} + 1429352290 p^{2} T^{12} - 13410012 p^{4} T^{14} + 92093 p^{6} T^{16} - 422 p^{8} T^{18} + p^{10} T^{20} \) | |
| 73 | \( 1 - 3 T + 115 T^{2} - 336 T^{3} + 8912 T^{4} - 58260 T^{5} + 290497 T^{6} - 1508853 T^{7} - 7279937 T^{8} + 55999116 T^{9} - 744209196 T^{10} + 55999116 p T^{11} - 7279937 p^{2} T^{12} - 1508853 p^{3} T^{13} + 290497 p^{4} T^{14} - 58260 p^{5} T^{15} + 8912 p^{6} T^{16} - 336 p^{7} T^{17} + 115 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - T - 201 T^{2} + 740 T^{3} + 18128 T^{4} - 100900 T^{5} - 805347 T^{6} + 5115611 T^{7} + 21370651 T^{8} - 77003688 T^{9} - 699661200 T^{10} - 77003688 p T^{11} + 21370651 p^{2} T^{12} + 5115611 p^{3} T^{13} - 805347 p^{4} T^{14} - 100900 p^{5} T^{15} + 18128 p^{6} T^{16} + 740 p^{7} T^{17} - 201 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( ( 1 + 4 T + 36 T^{2} + 304 T^{3} + 3029 T^{4} - 33442 T^{5} + 3029 p T^{6} + 304 p^{2} T^{7} + 36 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 89 | \( 1 + 28 T + 118 T^{2} - 2240 T^{3} + 12999 T^{4} + 7142 p T^{5} + 2053984 T^{6} - 15180164 T^{7} + 276922069 T^{8} + 3387791550 T^{9} + 10662822610 T^{10} + 3387791550 p T^{11} + 276922069 p^{2} T^{12} - 15180164 p^{3} T^{13} + 2053984 p^{4} T^{14} + 7142 p^{6} T^{15} + 12999 p^{6} T^{16} - 2240 p^{7} T^{17} + 118 p^{8} T^{18} + 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 758 T^{2} + 269933 T^{4} - 59888792 T^{6} + 9235543570 T^{8} - 1040392525668 T^{10} + 9235543570 p^{2} T^{12} - 59888792 p^{4} T^{14} + 269933 p^{6} T^{16} - 758 p^{8} T^{18} + p^{10} T^{20} \) | |
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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
−3.12900720538704315368915543583, −3.07986244032309568858054137574, −2.95534566755713612874058343220, −2.94146553488230531354203487022, −2.85448983247523262150534131584, −2.68044284239223477409472635921, −2.64596227467788769030280640255, −2.58027628156395078705437551006, −2.26221997484949477555477784321, −2.26066896724102259497551479221, −2.19099450778344596719841289235, −2.18288078549985607168031452429, −2.01708572460236632740536121028, −1.56422457806004392177810377402, −1.49916067658124262265747175643, −1.42322187227919366406661543850, −1.31920536836106230953724010478, −1.27898635375639981719830790977, −1.15413492647051049010179299560, −1.12958112266404159432763636169, −0.934349205220245149337201031230, −0.78629608517346190130845661429, −0.68737450260328778298243226623, −0.47663054593007938500506535779, −0.41242080966833122927957889672, 0.41242080966833122927957889672, 0.47663054593007938500506535779, 0.68737450260328778298243226623, 0.78629608517346190130845661429, 0.934349205220245149337201031230, 1.12958112266404159432763636169, 1.15413492647051049010179299560, 1.27898635375639981719830790977, 1.31920536836106230953724010478, 1.42322187227919366406661543850, 1.49916067658124262265747175643, 1.56422457806004392177810377402, 2.01708572460236632740536121028, 2.18288078549985607168031452429, 2.19099450778344596719841289235, 2.26066896724102259497551479221, 2.26221997484949477555477784321, 2.58027628156395078705437551006, 2.64596227467788769030280640255, 2.68044284239223477409472635921, 2.85448983247523262150534131584, 2.94146553488230531354203487022, 2.95534566755713612874058343220, 3.07986244032309568858054137574, 3.12900720538704315368915543583
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.