Dirichlet series
| L(s) = 1 | + 10·2-s + 5·3-s + 55·4-s − 5-s + 50·6-s + 2·7-s + 220·8-s + 3·9-s − 10·10-s + 11-s + 275·12-s + 9·13-s + 20·14-s − 5·15-s + 715·16-s + 30·18-s + 20·19-s − 55·20-s + 10·21-s + 10·22-s + 10·23-s + 1.10e3·24-s − 13·25-s + 90·26-s − 24·27-s + 110·28-s + 10·29-s + ⋯ |
| L(s) = 1 | + 7.07·2-s + 2.88·3-s + 55/2·4-s − 0.447·5-s + 20.4·6-s + 0.755·7-s + 77.7·8-s + 9-s − 3.16·10-s + 0.301·11-s + 79.3·12-s + 2.49·13-s + 5.34·14-s − 1.29·15-s + 178.·16-s + 7.07·18-s + 4.58·19-s − 12.2·20-s + 2.18·21-s + 2.13·22-s + 2.08·23-s + 224.·24-s − 2.59·25-s + 17.6·26-s − 4.61·27-s + 20.7·28-s + 1.85·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 23^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 23^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 23^{10} \cdot 29^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.88074\times 10^{10}\) |
| Root analytic conductor: | \(3.26374\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 23^{10} \cdot 29^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(14407.23885\) |
| \(L(\frac12)\) | \(\approx\) | \(14407.23885\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{10} \) |
| 23 | \( ( 1 - T )^{10} \) | |
| 29 | \( ( 1 - T )^{10} \) | |
| good | 3 | \( 1 - 5 T + 22 T^{2} - 71 T^{3} + 212 T^{4} - 547 T^{5} + 1322 T^{6} - 2881 T^{7} + 5947 T^{8} - 11312 T^{9} + 20368 T^{10} - 11312 p T^{11} + 5947 p^{2} T^{12} - 2881 p^{3} T^{13} + 1322 p^{4} T^{14} - 547 p^{5} T^{15} + 212 p^{6} T^{16} - 71 p^{7} T^{17} + 22 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 + T + 14 T^{2} + 29 T^{3} + 118 T^{4} + 221 T^{5} + 848 T^{6} + 199 p T^{7} + 169 p^{2} T^{8} + 4938 T^{9} + 18584 T^{10} + 4938 p T^{11} + 169 p^{4} T^{12} + 199 p^{4} T^{13} + 848 p^{4} T^{14} + 221 p^{5} T^{15} + 118 p^{6} T^{16} + 29 p^{7} T^{17} + 14 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 - 2 T + 4 p T^{2} - 46 T^{3} + 435 T^{4} - 2 p^{3} T^{5} + 732 p T^{6} - 7578 T^{7} + 48156 T^{8} - 63928 T^{9} + 368832 T^{10} - 63928 p T^{11} + 48156 p^{2} T^{12} - 7578 p^{3} T^{13} + 732 p^{5} T^{14} - 2 p^{8} T^{15} + 435 p^{6} T^{16} - 46 p^{7} T^{17} + 4 p^{9} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - T + 54 T^{2} - 7 T^{3} + 1454 T^{4} + 65 p T^{5} + 27792 T^{6} + 20349 T^{7} + 38715 p T^{8} + 303108 T^{9} + 5254500 T^{10} + 303108 p T^{11} + 38715 p^{3} T^{12} + 20349 p^{3} T^{13} + 27792 p^{4} T^{14} + 65 p^{6} T^{15} + 1454 p^{6} T^{16} - 7 p^{7} T^{17} + 54 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 9 T + 8 p T^{2} - 645 T^{3} + 4336 T^{4} - 20357 T^{5} + 101216 T^{6} - 384613 T^{7} + 1602231 T^{8} - 5404816 T^{9} + 21302848 T^{10} - 5404816 p T^{11} + 1602231 p^{2} T^{12} - 384613 p^{3} T^{13} + 101216 p^{4} T^{14} - 20357 p^{5} T^{15} + 4336 p^{6} T^{16} - 645 p^{7} T^{17} + 8 p^{9} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 108 T^{2} + 88 T^{3} + 5709 T^{4} + 7498 T^{5} + 692 p^{2} T^{6} + 296886 T^{7} + 5150266 T^{8} + 7310496 T^{9} + 100423736 T^{10} + 7310496 p T^{11} + 5150266 p^{2} T^{12} + 296886 p^{3} T^{13} + 692 p^{6} T^{14} + 7498 p^{5} T^{15} + 5709 p^{6} T^{16} + 88 p^{7} T^{17} + 108 p^{8} T^{18} + p^{10} T^{20} \) | |
| 19 | \( 1 - 20 T + 244 T^{2} - 2092 T^{3} + 14641 T^{4} - 88420 T^{5} + 494640 T^{6} - 2601052 T^{7} + 13051894 T^{8} - 61736432 T^{9} + 277226360 T^{10} - 61736432 p T^{11} + 13051894 p^{2} T^{12} - 2601052 p^{3} T^{13} + 494640 p^{4} T^{14} - 88420 p^{5} T^{15} + 14641 p^{6} T^{16} - 2092 p^{7} T^{17} + 244 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 21 T + 252 T^{2} - 2147 T^{3} + 15978 T^{4} - 3759 p T^{5} + 859178 T^{6} - 5889531 T^{7} + 36566725 T^{8} - 208157800 T^{9} + 1160266836 T^{10} - 208157800 p T^{11} + 36566725 p^{2} T^{12} - 5889531 p^{3} T^{13} + 859178 p^{4} T^{14} - 3759 p^{6} T^{15} + 15978 p^{6} T^{16} - 2147 p^{7} T^{17} + 252 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 8 T + 198 T^{2} + 1168 T^{3} + 19815 T^{4} + 101550 T^{5} + 1399700 T^{6} + 6354670 T^{7} + 74464032 T^{8} + 300567076 T^{9} + 3095384588 T^{10} + 300567076 p T^{11} + 74464032 p^{2} T^{12} + 6354670 p^{3} T^{13} + 1399700 p^{4} T^{14} + 101550 p^{5} T^{15} + 19815 p^{6} T^{16} + 1168 p^{7} T^{17} + 198 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 4 T + 146 T^{2} + 516 T^{3} + 12527 T^{4} + 46126 T^{5} + 806112 T^{6} + 2824918 T^{7} + 40950512 T^{8} + 141488708 T^{9} + 1810172860 T^{10} + 141488708 p T^{11} + 40950512 p^{2} T^{12} + 2824918 p^{3} T^{13} + 806112 p^{4} T^{14} + 46126 p^{5} T^{15} + 12527 p^{6} T^{16} + 516 p^{7} T^{17} + 146 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 3 T + 282 T^{2} + 881 T^{3} + 38242 T^{4} + 114523 T^{5} + 3345872 T^{6} + 9122373 T^{7} + 212250485 T^{8} + 516070768 T^{9} + 10327329692 T^{10} + 516070768 p T^{11} + 212250485 p^{2} T^{12} + 9122373 p^{3} T^{13} + 3345872 p^{4} T^{14} + 114523 p^{5} T^{15} + 38242 p^{6} T^{16} + 881 p^{7} T^{17} + 282 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + T + 338 T^{2} + 785 T^{3} + 52524 T^{4} + 188765 T^{5} + 5057938 T^{6} + 23254907 T^{7} + 346134675 T^{8} + 1720580030 T^{9} + 18257113144 T^{10} + 1720580030 p T^{11} + 346134675 p^{2} T^{12} + 23254907 p^{3} T^{13} + 5057938 p^{4} T^{14} + 188765 p^{5} T^{15} + 52524 p^{6} T^{16} + 785 p^{7} T^{17} + 338 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 13 T + 306 T^{2} + 2825 T^{3} + 37518 T^{4} + 249105 T^{5} + 2251004 T^{6} + 9052847 T^{7} + 57142577 T^{8} - 10440366 T^{9} + 613808808 T^{10} - 10440366 p T^{11} + 57142577 p^{2} T^{12} + 9052847 p^{3} T^{13} + 2251004 p^{4} T^{14} + 249105 p^{5} T^{15} + 37518 p^{6} T^{16} + 2825 p^{7} T^{17} + 306 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 14 T + 326 T^{2} - 3718 T^{3} + 56213 T^{4} - 555516 T^{5} + 6548968 T^{6} - 57000476 T^{7} + 567067010 T^{8} - 4382215472 T^{9} + 37878021348 T^{10} - 4382215472 p T^{11} + 567067010 p^{2} T^{12} - 57000476 p^{3} T^{13} + 6548968 p^{4} T^{14} - 555516 p^{5} T^{15} + 56213 p^{6} T^{16} - 3718 p^{7} T^{17} + 326 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 12 T + 474 T^{2} - 4908 T^{3} + 106587 T^{4} - 959970 T^{5} + 15009060 T^{6} - 117850402 T^{7} + 1468747836 T^{8} - 10013377004 T^{9} + 104434729988 T^{10} - 10013377004 p T^{11} + 1468747836 p^{2} T^{12} - 117850402 p^{3} T^{13} + 15009060 p^{4} T^{14} - 959970 p^{5} T^{15} + 106587 p^{6} T^{16} - 4908 p^{7} T^{17} + 474 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 6 T + 186 T^{2} - 8 T^{3} + 9013 T^{4} + 132256 T^{5} + 102028 T^{6} + 11503024 T^{7} + 60653554 T^{8} + 219424502 T^{9} + 8046289404 T^{10} + 219424502 p T^{11} + 60653554 p^{2} T^{12} + 11503024 p^{3} T^{13} + 102028 p^{4} T^{14} + 132256 p^{5} T^{15} + 9013 p^{6} T^{16} - 8 p^{7} T^{17} + 186 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 8 T + 330 T^{2} - 2976 T^{3} + 60669 T^{4} - 536056 T^{5} + 8078584 T^{6} - 65439800 T^{7} + 818514610 T^{8} - 6072638616 T^{9} + 65047188732 T^{10} - 6072638616 p T^{11} + 818514610 p^{2} T^{12} - 65439800 p^{3} T^{13} + 8078584 p^{4} T^{14} - 536056 p^{5} T^{15} + 60669 p^{6} T^{16} - 2976 p^{7} T^{17} + 330 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 16 T + 410 T^{2} - 4598 T^{3} + 69239 T^{4} - 654224 T^{5} + 7774984 T^{6} - 72425036 T^{7} + 757780200 T^{8} - 6895232430 T^{9} + 62492408028 T^{10} - 6895232430 p T^{11} + 757780200 p^{2} T^{12} - 72425036 p^{3} T^{13} + 7774984 p^{4} T^{14} - 654224 p^{5} T^{15} + 69239 p^{6} T^{16} - 4598 p^{7} T^{17} + 410 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 7 T + 464 T^{2} - 1159 T^{3} + 87184 T^{4} + 117973 T^{5} + 10902336 T^{6} + 35232053 T^{7} + 1249092739 T^{8} + 3113364576 T^{9} + 117252366580 T^{10} + 3113364576 p T^{11} + 1249092739 p^{2} T^{12} + 35232053 p^{3} T^{13} + 10902336 p^{4} T^{14} + 117973 p^{5} T^{15} + 87184 p^{6} T^{16} - 1159 p^{7} T^{17} + 464 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 18 T + 548 T^{2} - 7418 T^{3} + 126121 T^{4} - 1446392 T^{5} + 17732168 T^{6} - 186609968 T^{7} + 1842076158 T^{8} - 18561421288 T^{9} + 161546540744 T^{10} - 18561421288 p T^{11} + 1842076158 p^{2} T^{12} - 186609968 p^{3} T^{13} + 17732168 p^{4} T^{14} - 1446392 p^{5} T^{15} + 126121 p^{6} T^{16} - 7418 p^{7} T^{17} + 548 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 2 T + 416 T^{2} - 1228 T^{3} + 98393 T^{4} - 277156 T^{5} + 16228132 T^{6} - 41402516 T^{7} + 2030093470 T^{8} - 4628358914 T^{9} + 201423814480 T^{10} - 4628358914 p T^{11} + 2030093470 p^{2} T^{12} - 41402516 p^{3} T^{13} + 16228132 p^{4} T^{14} - 277156 p^{5} T^{15} + 98393 p^{6} T^{16} - 1228 p^{7} T^{17} + 416 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 6 T + 548 T^{2} - 1826 T^{3} + 129793 T^{4} - 97296 T^{5} + 18004376 T^{6} + 38732600 T^{7} + 1764439390 T^{8} + 8905083200 T^{9} + 160717023208 T^{10} + 8905083200 p T^{11} + 1764439390 p^{2} T^{12} + 38732600 p^{3} T^{13} + 18004376 p^{4} T^{14} - 97296 p^{5} T^{15} + 129793 p^{6} T^{16} - 1826 p^{7} T^{17} + 548 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.54206887420709801150167738463, −3.52806972895713462626475977991, −3.48279701582388317187927746422, −3.32608350969651380090376370879, −3.17330835425174528622979698540, −2.94866815839544048811111082102, −2.83670437235886265413095341948, −2.78326838191758894886642194260, −2.73747588511298536411500795183, −2.62247696977359216881977682237, −2.56986250941755044193579772037, −2.55811707414204038184582213038, −2.46711852886225520303473888375, −2.37995111271027038092236960792, −2.31641264547961800328191640595, −1.89942714688812303038230374742, −1.60160779234231749473945062082, −1.54828696934028570286393709503, −1.54725016103956798415865633780, −1.50550447042976015404119915262, −1.35736357551806557527178431027, −1.10408675346389531328816145294, −0.980527162513218390717693819316, −0.862845583331986098621430322804, −0.44544835577340504799579146045, 0.44544835577340504799579146045, 0.862845583331986098621430322804, 0.980527162513218390717693819316, 1.10408675346389531328816145294, 1.35736357551806557527178431027, 1.50550447042976015404119915262, 1.54725016103956798415865633780, 1.54828696934028570286393709503, 1.60160779234231749473945062082, 1.89942714688812303038230374742, 2.31641264547961800328191640595, 2.37995111271027038092236960792, 2.46711852886225520303473888375, 2.55811707414204038184582213038, 2.56986250941755044193579772037, 2.62247696977359216881977682237, 2.73747588511298536411500795183, 2.78326838191758894886642194260, 2.83670437235886265413095341948, 2.94866815839544048811111082102, 3.17330835425174528622979698540, 3.32608350969651380090376370879, 3.48279701582388317187927746422, 3.52806972895713462626475977991, 3.54206887420709801150167738463
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.