Dirichlet series
| L(s) = 1 | − 10·2-s − 10·3-s + 55·4-s + 5-s + 100·6-s + 10·7-s − 220·8-s + 55·9-s − 10·10-s + 2·11-s − 550·12-s − 100·14-s − 10·15-s + 715·16-s + 10·17-s − 550·18-s + 15·19-s + 55·20-s − 100·21-s − 20·22-s + 19·23-s + 2.20e3·24-s − 15·25-s − 220·27-s + 550·28-s − 29-s + 100·30-s + ⋯ |
| L(s) = 1 | − 7.07·2-s − 5.77·3-s + 55/2·4-s + 0.447·5-s + 40.8·6-s + 3.77·7-s − 77.7·8-s + 55/3·9-s − 3.16·10-s + 0.603·11-s − 158.·12-s − 26.7·14-s − 2.58·15-s + 178.·16-s + 2.42·17-s − 129.·18-s + 3.44·19-s + 12.2·20-s − 21.8·21-s − 4.26·22-s + 3.96·23-s + 449.·24-s − 3·25-s − 42.3·27-s + 103.·28-s − 0.185·29-s + 18.2·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 17^{10} \cdot 59^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 17^{10} \cdot 59^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{10} \cdot 17^{10} \cdot 59^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.56587\times 10^{16}\) |
| Root analytic conductor: | \(6.93209\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{10} \cdot 17^{10} \cdot 59^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.261569073\) |
| \(L(\frac12)\) | \(\approx\) | \(2.261569073\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{10} \) |
| 3 | \( ( 1 + T )^{10} \) | |
| 17 | \( ( 1 - T )^{10} \) | |
| 59 | \( ( 1 + T )^{10} \) | |
| good | 5 | \( 1 - T + 16 T^{2} - 3 p T^{3} + 106 T^{4} - 126 T^{5} + 398 T^{6} - 474 T^{7} + 901 T^{8} + 244 T^{9} + 2132 T^{10} + 244 p T^{11} + 901 p^{2} T^{12} - 474 p^{3} T^{13} + 398 p^{4} T^{14} - 126 p^{5} T^{15} + 106 p^{6} T^{16} - 3 p^{8} T^{17} + 16 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 - 10 T + 75 T^{2} - 418 T^{3} + 41 p^{2} T^{4} - 8287 T^{5} + 30980 T^{6} - 104534 T^{7} + 328204 T^{8} - 953199 T^{9} + 2612702 T^{10} - 953199 p T^{11} + 328204 p^{2} T^{12} - 104534 p^{3} T^{13} + 30980 p^{4} T^{14} - 8287 p^{5} T^{15} + 41 p^{8} T^{16} - 418 p^{7} T^{17} + 75 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 2 T + 46 T^{2} + 5 T^{3} + 940 T^{4} + 1701 T^{5} + 15490 T^{6} + 42545 T^{7} + 209859 T^{8} + 715659 T^{9} + 2346368 T^{10} + 715659 p T^{11} + 209859 p^{2} T^{12} + 42545 p^{3} T^{13} + 15490 p^{4} T^{14} + 1701 p^{5} T^{15} + 940 p^{6} T^{16} + 5 p^{7} T^{17} + 46 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 3 p T^{2} + 82 T^{3} + 968 T^{4} + 192 p T^{5} + 23358 T^{6} + 46212 T^{7} + 405807 T^{8} + 870802 T^{9} + 5459510 T^{10} + 870802 p T^{11} + 405807 p^{2} T^{12} + 46212 p^{3} T^{13} + 23358 p^{4} T^{14} + 192 p^{6} T^{15} + 968 p^{6} T^{16} + 82 p^{7} T^{17} + 3 p^{9} T^{18} + p^{10} T^{20} \) | |
| 19 | \( 1 - 15 T + 199 T^{2} - 1652 T^{3} + 12796 T^{4} - 76620 T^{5} + 457933 T^{6} - 2320711 T^{7} + 12250563 T^{8} - 56006184 T^{9} + 263591800 T^{10} - 56006184 p T^{11} + 12250563 p^{2} T^{12} - 2320711 p^{3} T^{13} + 457933 p^{4} T^{14} - 76620 p^{5} T^{15} + 12796 p^{6} T^{16} - 1652 p^{7} T^{17} + 199 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 19 T + 13 p T^{2} - 3139 T^{3} + 28833 T^{4} - 211264 T^{5} + 1405874 T^{6} - 7996629 T^{7} + 43279532 T^{8} - 212465565 T^{9} + 1050593858 T^{10} - 212465565 p T^{11} + 43279532 p^{2} T^{12} - 7996629 p^{3} T^{13} + 1405874 p^{4} T^{14} - 211264 p^{5} T^{15} + 28833 p^{6} T^{16} - 3139 p^{7} T^{17} + 13 p^{9} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + T + 152 T^{2} + 341 T^{3} + 11852 T^{4} + 38638 T^{5} + 633370 T^{6} + 2412694 T^{7} + 25925483 T^{8} + 99142794 T^{9} + 841964060 T^{10} + 99142794 p T^{11} + 25925483 p^{2} T^{12} + 2412694 p^{3} T^{13} + 633370 p^{4} T^{14} + 38638 p^{5} T^{15} + 11852 p^{6} T^{16} + 341 p^{7} T^{17} + 152 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 15 T + 234 T^{2} - 2397 T^{3} + 21704 T^{4} - 161656 T^{5} + 1059798 T^{6} - 6063172 T^{7} + 32048335 T^{8} - 162507738 T^{9} + 868334960 T^{10} - 162507738 p T^{11} + 32048335 p^{2} T^{12} - 6063172 p^{3} T^{13} + 1059798 p^{4} T^{14} - 161656 p^{5} T^{15} + 21704 p^{6} T^{16} - 2397 p^{7} T^{17} + 234 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - T + 184 T^{2} - 149 T^{3} + 14718 T^{4} - 10974 T^{5} + 667218 T^{6} - 592030 T^{7} + 19762489 T^{8} - 26578394 T^{9} + 575597964 T^{10} - 26578394 p T^{11} + 19762489 p^{2} T^{12} - 592030 p^{3} T^{13} + 667218 p^{4} T^{14} - 10974 p^{5} T^{15} + 14718 p^{6} T^{16} - 149 p^{7} T^{17} + 184 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 5 T + 325 T^{2} + 37 p T^{3} + 1221 p T^{4} + 214156 T^{5} + 4814010 T^{6} + 18501559 T^{7} + 319916256 T^{8} + 1080481419 T^{9} + 15343917382 T^{10} + 1080481419 p T^{11} + 319916256 p^{2} T^{12} + 18501559 p^{3} T^{13} + 4814010 p^{4} T^{14} + 214156 p^{5} T^{15} + 1221 p^{7} T^{16} + 37 p^{8} T^{17} + 325 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 26 T + 512 T^{2} - 6855 T^{3} + 80190 T^{4} - 766511 T^{5} + 6782614 T^{6} - 52375395 T^{7} + 389955937 T^{8} - 2646857257 T^{9} + 17977206388 T^{10} - 2646857257 p T^{11} + 389955937 p^{2} T^{12} - 52375395 p^{3} T^{13} + 6782614 p^{4} T^{14} - 766511 p^{5} T^{15} + 80190 p^{6} T^{16} - 6855 p^{7} T^{17} + 512 p^{8} T^{18} - 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 14 T + 348 T^{2} - 3667 T^{3} + 54992 T^{4} - 484131 T^{5} + 5561478 T^{6} - 42351223 T^{7} + 402147183 T^{8} - 2671469577 T^{9} + 21718582076 T^{10} - 2671469577 p T^{11} + 402147183 p^{2} T^{12} - 42351223 p^{3} T^{13} + 5561478 p^{4} T^{14} - 484131 p^{5} T^{15} + 54992 p^{6} T^{16} - 3667 p^{7} T^{17} + 348 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 2 T + 405 T^{2} + 926 T^{3} + 78481 T^{4} + 186453 T^{5} + 9575990 T^{6} + 22056540 T^{7} + 813784196 T^{8} + 1710472535 T^{9} + 50276749610 T^{10} + 1710472535 p T^{11} + 813784196 p^{2} T^{12} + 22056540 p^{3} T^{13} + 9575990 p^{4} T^{14} + 186453 p^{5} T^{15} + 78481 p^{6} T^{16} + 926 p^{7} T^{17} + 405 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 4 T + 478 T^{2} - 1863 T^{3} + 108950 T^{4} - 399955 T^{5} + 15556186 T^{6} - 52174469 T^{7} + 1537278929 T^{8} - 4568319325 T^{9} + 109690163744 T^{10} - 4568319325 p T^{11} + 1537278929 p^{2} T^{12} - 52174469 p^{3} T^{13} + 15556186 p^{4} T^{14} - 399955 p^{5} T^{15} + 108950 p^{6} T^{16} - 1863 p^{7} T^{17} + 478 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 15 T + 429 T^{2} - 5446 T^{3} + 91406 T^{4} - 1000859 T^{5} + 12769198 T^{6} - 121934859 T^{7} + 1290952545 T^{8} - 10813832283 T^{9} + 98756472682 T^{10} - 10813832283 p T^{11} + 1290952545 p^{2} T^{12} - 121934859 p^{3} T^{13} + 12769198 p^{4} T^{14} - 1000859 p^{5} T^{15} + 91406 p^{6} T^{16} - 5446 p^{7} T^{17} + 429 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 14 T + 480 T^{2} - 6135 T^{3} + 111962 T^{4} - 1278779 T^{5} + 17030166 T^{6} - 169954891 T^{7} + 1869416669 T^{8} - 16137639849 T^{9} + 153117235508 T^{10} - 16137639849 p T^{11} + 1869416669 p^{2} T^{12} - 169954891 p^{3} T^{13} + 17030166 p^{4} T^{14} - 1278779 p^{5} T^{15} + 111962 p^{6} T^{16} - 6135 p^{7} T^{17} + 480 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 43 T + 1265 T^{2} - 26511 T^{3} + 461617 T^{4} - 6754708 T^{5} + 87998606 T^{6} - 1021936865 T^{7} + 10878475704 T^{8} - 105354721041 T^{9} + 942066932846 T^{10} - 105354721041 p T^{11} + 10878475704 p^{2} T^{12} - 1021936865 p^{3} T^{13} + 87998606 p^{4} T^{14} - 6754708 p^{5} T^{15} + 461617 p^{6} T^{16} - 26511 p^{7} T^{17} + 1265 p^{8} T^{18} - 43 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 428 T^{2} + 28 T^{3} + 97162 T^{4} + 852 T^{5} + 15015246 T^{6} + 209960 T^{7} + 1729283261 T^{8} + 36203184 T^{9} + 154471209180 T^{10} + 36203184 p T^{11} + 1729283261 p^{2} T^{12} + 209960 p^{3} T^{13} + 15015246 p^{4} T^{14} + 852 p^{5} T^{15} + 97162 p^{6} T^{16} + 28 p^{7} T^{17} + 428 p^{8} T^{18} + p^{10} T^{20} \) | |
| 83 | \( 1 + 4 T + 454 T^{2} + 1670 T^{3} + 98929 T^{4} + 340815 T^{5} + 14368903 T^{6} + 44475310 T^{7} + 1598660102 T^{8} + 4310958037 T^{9} + 145321537494 T^{10} + 4310958037 p T^{11} + 1598660102 p^{2} T^{12} + 44475310 p^{3} T^{13} + 14368903 p^{4} T^{14} + 340815 p^{5} T^{15} + 98929 p^{6} T^{16} + 1670 p^{7} T^{17} + 454 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 22 T + 768 T^{2} + 12056 T^{3} + 247577 T^{4} + 3070135 T^{5} + 47437575 T^{6} + 491580966 T^{7} + 6308467804 T^{8} + 56747582645 T^{9} + 635334397110 T^{10} + 56747582645 p T^{11} + 6308467804 p^{2} T^{12} + 491580966 p^{3} T^{13} + 47437575 p^{4} T^{14} + 3070135 p^{5} T^{15} + 247577 p^{6} T^{16} + 12056 p^{7} T^{17} + 768 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 37 T + 1088 T^{2} - 22621 T^{3} + 400557 T^{4} - 5971412 T^{5} + 79694671 T^{6} - 955015727 T^{7} + 10701319044 T^{8} - 112000135835 T^{9} + 1132020155870 T^{10} - 112000135835 p T^{11} + 10701319044 p^{2} T^{12} - 955015727 p^{3} T^{13} + 79694671 p^{4} T^{14} - 5971412 p^{5} T^{15} + 400557 p^{6} T^{16} - 22621 p^{7} T^{17} + 1088 p^{8} T^{18} - 37 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−2.54103942865400453659096777168, −2.47531346497892199187740782135, −2.43013663290177664421332344910, −2.39456772760247263183441787958, −2.37080920451915986394013477281, −1.82274270909881381136079085453, −1.82046215184681348639973338087, −1.77643119847789249150777787596, −1.61152737186255826439554059865, −1.57297546666798783402659700765, −1.54644615063061868542741810867, −1.53227823355358950216201158173, −1.53186868896680664083990763998, −1.47155375206080000266496355911, −1.19973792873057650811596414198, −1.18835383015071006714047264235, −0.936334764565943426754488473594, −0.794875533242016920508969725230, −0.77482212046702281966343039181, −0.65478290436927012214256685895, −0.59307268374123515490891182617, −0.57538661755357684507457007619, −0.51791595051082262050846467837, −0.49285769817402343732726080945, −0.46508801446345660511828806478, 0.46508801446345660511828806478, 0.49285769817402343732726080945, 0.51791595051082262050846467837, 0.57538661755357684507457007619, 0.59307268374123515490891182617, 0.65478290436927012214256685895, 0.77482212046702281966343039181, 0.794875533242016920508969725230, 0.936334764565943426754488473594, 1.18835383015071006714047264235, 1.19973792873057650811596414198, 1.47155375206080000266496355911, 1.53186868896680664083990763998, 1.53227823355358950216201158173, 1.54644615063061868542741810867, 1.57297546666798783402659700765, 1.61152737186255826439554059865, 1.77643119847789249150777787596, 1.82046215184681348639973338087, 1.82274270909881381136079085453, 2.37080920451915986394013477281, 2.39456772760247263183441787958, 2.43013663290177664421332344910, 2.47531346497892199187740782135, 2.54103942865400453659096777168
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.