Dirichlet series
| L(s) = 1 | + 2-s + 5·3-s + 8·5-s + 5·6-s + 7-s + 14·9-s + 8·10-s + 11·11-s − 6·13-s + 14-s + 40·15-s − 2·17-s + 14·18-s − 3·19-s + 5·21-s + 11·22-s − 21·23-s + 27·25-s − 6·26-s + 22·27-s + 2·29-s + 40·30-s + 3·31-s + 55·33-s − 2·34-s + 8·35-s − 6·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.88·3-s + 3.57·5-s + 2.04·6-s + 0.377·7-s + 14/3·9-s + 2.52·10-s + 3.31·11-s − 1.66·13-s + 0.267·14-s + 10.3·15-s − 0.485·17-s + 3.29·18-s − 0.688·19-s + 1.09·21-s + 2.34·22-s − 4.37·23-s + 27/5·25-s − 1.17·26-s + 4.23·27-s + 0.371·29-s + 7.30·30-s + 0.538·31-s + 9.57·33-s − 0.342·34-s + 1.35·35-s − 0.986·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 7^{10} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(12627.8\) |
| Root analytic conductor: | \(1.60349\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 7^{10} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(34.10056019\) |
| \(L(\frac12)\) | \(\approx\) | \(34.10056019\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 7 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) | |
| 23 | \( 1 + 21 T + 177 T^{2} + 505 T^{3} - 2705 T^{4} - 27325 T^{5} - 2705 p T^{6} + 505 p^{2} T^{7} + 177 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 3 | \( 1 - 5 T + 11 T^{2} - 7 T^{3} - p^{2} T^{4} + 11 T^{5} - 2 p T^{6} + 43 p T^{7} - 110 p T^{8} + 130 T^{9} + 439 T^{10} + 130 p T^{11} - 110 p^{3} T^{12} + 43 p^{4} T^{13} - 2 p^{5} T^{14} + 11 p^{5} T^{15} - p^{8} T^{16} - 7 p^{7} T^{17} + 11 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 - 8 T + 37 T^{2} - 113 T^{3} + 47 p T^{4} - 226 T^{5} - 67 p T^{6} + 399 p T^{7} - 844 p T^{8} + 5404 T^{9} - 6039 T^{10} + 5404 p T^{11} - 844 p^{3} T^{12} + 399 p^{4} T^{13} - 67 p^{5} T^{14} - 226 p^{5} T^{15} + 47 p^{7} T^{16} - 113 p^{7} T^{17} + 37 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - p T + 4 p T^{2} - 10 p T^{3} + 24 p T^{4} + 13 p T^{5} - 12 p^{2} T^{6} - 68 p^{2} T^{7} + 382 p^{2} T^{8} - 2174 p^{2} T^{9} + 10671 p^{2} T^{10} - 2174 p^{3} T^{11} + 382 p^{4} T^{12} - 68 p^{5} T^{13} - 12 p^{6} T^{14} + 13 p^{6} T^{15} + 24 p^{7} T^{16} - 10 p^{8} T^{17} + 4 p^{9} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 6 T + T^{2} - 138 T^{3} - 489 T^{4} + 15 T^{5} + 7019 T^{6} + 2342 p T^{7} + 53347 T^{8} - 303845 T^{9} - 1987569 T^{10} - 303845 p T^{11} + 53347 p^{2} T^{12} + 2342 p^{4} T^{13} + 7019 p^{4} T^{14} + 15 p^{5} T^{15} - 489 p^{6} T^{16} - 138 p^{7} T^{17} + p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 2 T - 13 T^{2} - 10 p T^{3} + 343 T^{4} + 4632 T^{5} + 13817 T^{6} - 88136 T^{7} - 361243 T^{8} + 576770 T^{9} + 10463331 T^{10} + 576770 p T^{11} - 361243 p^{2} T^{12} - 88136 p^{3} T^{13} + 13817 p^{4} T^{14} + 4632 p^{5} T^{15} + 343 p^{6} T^{16} - 10 p^{8} T^{17} - 13 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 3 T + 12 T^{2} + 7 p T^{3} + 1073 T^{4} + 1924 T^{5} + 875 p T^{6} + 87855 T^{7} + 399614 T^{8} + 1105831 T^{9} + 8422853 T^{10} + 1105831 p T^{11} + 399614 p^{2} T^{12} + 87855 p^{3} T^{13} + 875 p^{5} T^{14} + 1924 p^{5} T^{15} + 1073 p^{6} T^{16} + 7 p^{8} T^{17} + 12 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 2 T - 14 T^{2} + 350 T^{3} + 564 T^{4} - 9364 T^{5} + 55117 T^{6} + 437246 T^{7} - 1597978 T^{8} - 1533488 T^{9} + 105700849 T^{10} - 1533488 p T^{11} - 1597978 p^{2} T^{12} + 437246 p^{3} T^{13} + 55117 p^{4} T^{14} - 9364 p^{5} T^{15} + 564 p^{6} T^{16} + 350 p^{7} T^{17} - 14 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 3 T - 11 T^{2} - 149 T^{3} + 788 T^{4} - 3707 T^{5} + 62230 T^{6} - 109019 T^{7} + 1071873 T^{8} - 8975622 T^{9} + 23628549 T^{10} - 8975622 p T^{11} + 1071873 p^{2} T^{12} - 109019 p^{3} T^{13} + 62230 p^{4} T^{14} - 3707 p^{5} T^{15} + 788 p^{6} T^{16} - 149 p^{7} T^{17} - 11 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 6 T - 67 T^{2} - 844 T^{3} + 11 T^{4} + 40204 T^{5} + 170395 T^{6} - 998502 T^{7} - 8279439 T^{8} + 9144630 T^{9} + 309441067 T^{10} + 9144630 p T^{11} - 8279439 p^{2} T^{12} - 998502 p^{3} T^{13} + 170395 p^{4} T^{14} + 40204 p^{5} T^{15} + 11 p^{6} T^{16} - 844 p^{7} T^{17} - 67 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 23 T + 180 T^{2} + 458 T^{3} + 998 T^{4} + 38243 T^{5} + 439118 T^{6} + 2230016 T^{7} + 6359344 T^{8} + 45800880 T^{9} + 432910367 T^{10} + 45800880 p T^{11} + 6359344 p^{2} T^{12} + 2230016 p^{3} T^{13} + 439118 p^{4} T^{14} + 38243 p^{5} T^{15} + 998 p^{6} T^{16} + 458 p^{7} T^{17} + 180 p^{8} T^{18} + 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + T - 75 T^{2} - 778 T^{3} + 3063 T^{4} + 58847 T^{5} + 139988 T^{6} - 2958715 T^{7} - 20999956 T^{8} + 61258109 T^{9} + 1224617571 T^{10} + 61258109 p T^{11} - 20999956 p^{2} T^{12} - 2958715 p^{3} T^{13} + 139988 p^{4} T^{14} + 58847 p^{5} T^{15} + 3063 p^{6} T^{16} - 778 p^{7} T^{17} - 75 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 - 19 T + 320 T^{2} - 3279 T^{3} + 31293 T^{4} - 219641 T^{5} + 31293 p T^{6} - 3279 p^{2} T^{7} + 320 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 + 49 T + 1072 T^{2} + 13961 T^{3} + 128005 T^{4} + 1038086 T^{5} + 9283669 T^{6} + 86045833 T^{7} + 703289632 T^{8} + 5024178141 T^{9} + 35332526955 T^{10} + 5024178141 p T^{11} + 703289632 p^{2} T^{12} + 86045833 p^{3} T^{13} + 9283669 p^{4} T^{14} + 1038086 p^{5} T^{15} + 128005 p^{6} T^{16} + 13961 p^{7} T^{17} + 1072 p^{8} T^{18} + 49 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 9 T - 55 T^{2} + 597 T^{3} + 2327 T^{4} + 1639 T^{5} - 215844 T^{6} - 1317881 T^{7} + 18169470 T^{8} + 59198100 T^{9} - 1351247921 T^{10} + 59198100 p T^{11} + 18169470 p^{2} T^{12} - 1317881 p^{3} T^{13} - 215844 p^{4} T^{14} + 1639 p^{5} T^{15} + 2327 p^{6} T^{16} + 597 p^{7} T^{17} - 55 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 16 T - 47 T^{2} + 1904 T^{3} - 5839 T^{4} - 77346 T^{5} + 698139 T^{6} + 2027914 T^{7} - 56670385 T^{8} - 62456966 T^{9} + 4229997883 T^{10} - 62456966 p T^{11} - 56670385 p^{2} T^{12} + 2027914 p^{3} T^{13} + 698139 p^{4} T^{14} - 77346 p^{5} T^{15} - 5839 p^{6} T^{16} + 1904 p^{7} T^{17} - 47 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 7 T + 147 T^{2} - 1253 T^{3} + 18524 T^{4} - 145476 T^{5} + 1781776 T^{6} - 13496113 T^{7} + 150966936 T^{8} - 1084921013 T^{9} + 10014792833 T^{10} - 1084921013 p T^{11} + 150966936 p^{2} T^{12} - 13496113 p^{3} T^{13} + 1781776 p^{4} T^{14} - 145476 p^{5} T^{15} + 18524 p^{6} T^{16} - 1253 p^{7} T^{17} + 147 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 16 T - 123 T^{2} - 3104 T^{3} + 6600 T^{4} + 283788 T^{5} - 397886 T^{6} - 11948154 T^{7} + 102872926 T^{8} + 245745984 T^{9} - 10591376267 T^{10} + 245745984 p T^{11} + 102872926 p^{2} T^{12} - 11948154 p^{3} T^{13} - 397886 p^{4} T^{14} + 283788 p^{5} T^{15} + 6600 p^{6} T^{16} - 3104 p^{7} T^{17} - 123 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 9 T + 195 T^{2} + 2429 T^{3} + 20793 T^{4} + 334991 T^{5} + 2405784 T^{6} + 29659505 T^{7} + 274948622 T^{8} + 1918549512 T^{9} + 23619429175 T^{10} + 1918549512 p T^{11} + 274948622 p^{2} T^{12} + 29659505 p^{3} T^{13} + 2405784 p^{4} T^{14} + 334991 p^{5} T^{15} + 20793 p^{6} T^{16} + 2429 p^{7} T^{17} + 195 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 6 T + 23 T^{2} + 666 T^{3} + 1733 T^{4} + 17134 T^{5} + 829797 T^{6} - 4004460 T^{7} + 78382169 T^{8} + 343923374 T^{9} - 235850209 T^{10} + 343923374 p T^{11} + 78382169 p^{2} T^{12} - 4004460 p^{3} T^{13} + 829797 p^{4} T^{14} + 17134 p^{5} T^{15} + 1733 p^{6} T^{16} + 666 p^{7} T^{17} + 23 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 19 T + 146 T^{2} + 834 T^{3} + 2936 T^{4} + 1940 T^{5} + 44566 T^{6} - 1061154 T^{7} + 13933831 T^{8} + 829265641 T^{9} + 11267235440 T^{10} + 829265641 p T^{11} + 13933831 p^{2} T^{12} - 1061154 p^{3} T^{13} + 44566 p^{4} T^{14} + 1940 p^{5} T^{15} + 2936 p^{6} T^{16} + 834 p^{7} T^{17} + 146 p^{8} T^{18} + 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 25 T + 415 T^{2} - 5565 T^{3} + 64768 T^{4} - 546558 T^{5} + 2985898 T^{6} - 1630373 T^{7} - 260082868 T^{8} + 4542109341 T^{9} - 51638753099 T^{10} + 4542109341 p T^{11} - 260082868 p^{2} T^{12} - 1630373 p^{3} T^{13} + 2985898 p^{4} T^{14} - 546558 p^{5} T^{15} + 64768 p^{6} T^{16} - 5565 p^{7} T^{17} + 415 p^{8} T^{18} - 25 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 16 T - 72 T^{2} + 4354 T^{3} - 22057 T^{4} - 486513 T^{5} + 5296939 T^{6} + 38344565 T^{7} - 750080187 T^{8} - 1498239824 T^{9} + 85630134557 T^{10} - 1498239824 p T^{11} - 750080187 p^{2} T^{12} + 38344565 p^{3} T^{13} + 5296939 p^{4} T^{14} - 486513 p^{5} T^{15} - 22057 p^{6} T^{16} + 4354 p^{7} T^{17} - 72 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−4.53674861836790713676030725887, −4.37810196037198510988641885324, −4.18317770780819553232530768722, −3.95486319032537801972084302317, −3.89203799730654321833026863603, −3.87898722201574075842081127582, −3.84302416351565754098837456511, −3.52131828127895242989254600776, −3.41917337816038608725090359688, −3.34985778389422224737299907686, −3.05998613223561165622417772253, −2.99753176627521018734968423036, −2.90654171627731191677930709743, −2.69408108936262065799204604912, −2.44583502382680616283163680809, −2.19424205659108371975415240503, −2.15583839775746401128535120499, −2.09232341369519639356994325577, −1.89935566022407471148334666280, −1.84806010897408607257567588641, −1.79882610889272672425652570378, −1.59666220529050133095844519986, −1.40176430173687454086429756090, −1.17859864469636960180561333639, −0.47535222644247583012206415798, 0.47535222644247583012206415798, 1.17859864469636960180561333639, 1.40176430173687454086429756090, 1.59666220529050133095844519986, 1.79882610889272672425652570378, 1.84806010897408607257567588641, 1.89935566022407471148334666280, 2.09232341369519639356994325577, 2.15583839775746401128535120499, 2.19424205659108371975415240503, 2.44583502382680616283163680809, 2.69408108936262065799204604912, 2.90654171627731191677930709743, 2.99753176627521018734968423036, 3.05998613223561165622417772253, 3.34985778389422224737299907686, 3.41917337816038608725090359688, 3.52131828127895242989254600776, 3.84302416351565754098837456511, 3.87898722201574075842081127582, 3.89203799730654321833026863603, 3.95486319032537801972084302317, 4.18317770780819553232530768722, 4.37810196037198510988641885324, 4.53674861836790713676030725887
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.