Dirichlet series
| L(s) = 1 | − 2-s − 3-s + 8·5-s + 6-s + 8·7-s − 8·10-s + 7·11-s + 3·13-s − 8·14-s − 8·15-s + 4·17-s − 8·21-s − 7·22-s − 12·23-s + 27·25-s − 3·26-s − 25·29-s + 8·30-s + 6·31-s − 7·33-s − 4·34-s + 64·35-s + 9·37-s − 3·39-s + 24·41-s + 8·42-s − 30·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 3.57·5-s + 0.408·6-s + 3.02·7-s − 2.52·10-s + 2.11·11-s + 0.832·13-s − 2.13·14-s − 2.06·15-s + 0.970·17-s − 1.74·21-s − 1.49·22-s − 2.50·23-s + 27/5·25-s − 0.588·26-s − 4.64·29-s + 1.46·30-s + 1.07·31-s − 1.21·33-s − 0.685·34-s + 10.8·35-s + 1.47·37-s − 0.480·39-s + 3.74·41-s + 1.23·42-s − 4.57·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{10} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.63974\) |
| Root analytic conductor: | \(1.04973\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{10} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.594874246\) |
| \(L(\frac12)\) | \(\approx\) | \(2.594874246\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 3 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) | |
| 23 | \( 1 + 12 T - 10 T^{2} - 527 T^{3} - 32 T^{4} + 14103 T^{5} - 32 p T^{6} - 527 p^{2} T^{7} - 10 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 5 | \( 1 - 8 T + 37 T^{2} - 124 T^{3} + 334 T^{4} - 666 T^{5} + 732 T^{6} + 168 p T^{7} - 7586 T^{8} + 26854 T^{9} - 69091 T^{10} + 26854 p T^{11} - 7586 p^{2} T^{12} + 168 p^{4} T^{13} + 732 p^{4} T^{14} - 666 p^{5} T^{15} + 334 p^{6} T^{16} - 124 p^{7} T^{17} + 37 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 - 8 T + 24 T^{2} - 48 T^{3} + 34 p T^{4} - 1282 T^{5} + 4201 T^{6} - 9190 T^{7} + 22300 T^{8} - 89804 T^{9} + 301445 T^{10} - 89804 p T^{11} + 22300 p^{2} T^{12} - 9190 p^{3} T^{13} + 4201 p^{4} T^{14} - 1282 p^{5} T^{15} + 34 p^{7} T^{16} - 48 p^{7} T^{17} + 24 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 7 T - 6 T^{2} + 152 T^{3} - 118 T^{4} - 2430 T^{5} + 6538 T^{6} + 22456 T^{7} - 121915 T^{8} - 116091 T^{9} + 1687544 T^{10} - 116091 p T^{11} - 121915 p^{2} T^{12} + 22456 p^{3} T^{13} + 6538 p^{4} T^{14} - 2430 p^{5} T^{15} - 118 p^{6} T^{16} + 152 p^{7} T^{17} - 6 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 3 T + 18 T^{2} - 103 T^{3} + 647 T^{4} - 2956 T^{5} + 11897 T^{6} - 52549 T^{7} + 218696 T^{8} - 890703 T^{9} + 2888315 T^{10} - 890703 p T^{11} + 218696 p^{2} T^{12} - 52549 p^{3} T^{13} + 11897 p^{4} T^{14} - 2956 p^{5} T^{15} + 647 p^{6} T^{16} - 103 p^{7} T^{17} + 18 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 4 T - 12 T^{2} + 226 T^{3} - 227 T^{4} - 3385 T^{5} + 13703 T^{6} + 75399 T^{7} - 361957 T^{8} - 559372 T^{9} + 10194097 T^{10} - 559372 p T^{11} - 361957 p^{2} T^{12} + 75399 p^{3} T^{13} + 13703 p^{4} T^{14} - 3385 p^{5} T^{15} - 227 p^{6} T^{16} + 226 p^{7} T^{17} - 12 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 3 T^{2} - 44 T^{3} - 2 T^{4} - 484 T^{5} - 2206 T^{6} - 4994 T^{7} - 28772 T^{8} + 517286 T^{9} + 2075349 T^{10} + 517286 p T^{11} - 28772 p^{2} T^{12} - 4994 p^{3} T^{13} - 2206 p^{4} T^{14} - 484 p^{5} T^{15} - 2 p^{6} T^{16} - 44 p^{7} T^{17} + 3 p^{8} T^{18} + p^{10} T^{20} \) | |
| 29 | \( 1 + 25 T + 343 T^{2} + 3758 T^{3} + 36175 T^{4} + 307763 T^{5} + 2361028 T^{6} + 571586 p T^{7} + 107168251 T^{8} + 644460300 T^{9} + 3605981457 T^{10} + 644460300 p T^{11} + 107168251 p^{2} T^{12} + 571586 p^{4} T^{13} + 2361028 p^{4} T^{14} + 307763 p^{5} T^{15} + 36175 p^{6} T^{16} + 3758 p^{7} T^{17} + 343 p^{8} T^{18} + 25 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 6 T + 27 T^{2} - 262 T^{3} - 1256 T^{4} + 454 p T^{5} - 63086 T^{6} + 416630 T^{7} + 655194 T^{8} - 18053108 T^{9} + 103555045 T^{10} - 18053108 p T^{11} + 655194 p^{2} T^{12} + 416630 p^{3} T^{13} - 63086 p^{4} T^{14} + 454 p^{6} T^{15} - 1256 p^{6} T^{16} - 262 p^{7} T^{17} + 27 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 9 T - 11 T^{2} + 597 T^{3} - 1578 T^{4} - 17490 T^{5} + 70508 T^{6} + 726667 T^{7} - 5358452 T^{8} - 20203915 T^{9} + 314187213 T^{10} - 20203915 p T^{11} - 5358452 p^{2} T^{12} + 726667 p^{3} T^{13} + 70508 p^{4} T^{14} - 17490 p^{5} T^{15} - 1578 p^{6} T^{16} + 597 p^{7} T^{17} - 11 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 24 T + 260 T^{2} - 1505 T^{3} + 1315 T^{4} + 70515 T^{5} - 762281 T^{6} + 3662240 T^{7} + 3399191 T^{8} - 225020026 T^{9} + 2039308877 T^{10} - 225020026 p T^{11} + 3399191 p^{2} T^{12} + 3662240 p^{3} T^{13} - 762281 p^{4} T^{14} + 70515 p^{5} T^{15} + 1315 p^{6} T^{16} - 1505 p^{7} T^{17} + 260 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 30 T + 384 T^{2} + 2849 T^{3} + 18259 T^{4} + 167071 T^{5} + 1643555 T^{6} + 12781010 T^{7} + 84691531 T^{8} + 563761690 T^{9} + 3787773417 T^{10} + 563761690 p T^{11} + 84691531 p^{2} T^{12} + 12781010 p^{3} T^{13} + 1643555 p^{4} T^{14} + 167071 p^{5} T^{15} + 18259 p^{6} T^{16} + 2849 p^{7} T^{17} + 384 p^{8} T^{18} + 30 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 + 24 T + 318 T^{2} + 2992 T^{3} + 23413 T^{4} + 164567 T^{5} + 23413 p T^{6} + 2992 p^{2} T^{7} + 318 p^{3} T^{8} + 24 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 - 15 T + 18 T^{2} + 877 T^{3} - 4341 T^{4} + 26796 T^{5} - 467371 T^{6} + 1340329 T^{7} + 25779238 T^{8} - 153653535 T^{9} + 284961555 T^{10} - 153653535 p T^{11} + 25779238 p^{2} T^{12} + 1340329 p^{3} T^{13} - 467371 p^{4} T^{14} + 26796 p^{5} T^{15} - 4341 p^{6} T^{16} + 877 p^{7} T^{17} + 18 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 5 T - 78 T^{2} + 124 T^{3} + 99 T^{4} + 33285 T^{5} + 187918 T^{6} - 2056812 T^{7} - 7633706 T^{8} + 30950502 T^{9} + 265370689 T^{10} + 30950502 p T^{11} - 7633706 p^{2} T^{12} - 2056812 p^{3} T^{13} + 187918 p^{4} T^{14} + 33285 p^{5} T^{15} + 99 p^{6} T^{16} + 124 p^{7} T^{17} - 78 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 12 T + 72 T^{2} + 176 T^{3} - 10310 T^{4} + 50614 T^{5} + 145237 T^{6} - 4876234 T^{7} + 26679684 T^{8} + 194810596 T^{9} - 2657630207 T^{10} + 194810596 p T^{11} + 26679684 p^{2} T^{12} - 4876234 p^{3} T^{13} + 145237 p^{4} T^{14} + 50614 p^{5} T^{15} - 10310 p^{6} T^{16} + 176 p^{7} T^{17} + 72 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 18 T + 191 T^{2} - 1506 T^{3} + 9823 T^{4} - 98143 T^{5} + 981449 T^{6} - 9691752 T^{7} + 91079779 T^{8} - 824117789 T^{9} + 7576341785 T^{10} - 824117789 p T^{11} + 91079779 p^{2} T^{12} - 9691752 p^{3} T^{13} + 981449 p^{4} T^{14} - 98143 p^{5} T^{15} + 9823 p^{6} T^{16} - 1506 p^{7} T^{17} + 191 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 28 T + 317 T^{2} - 1300 T^{3} - 9691 T^{4} + 129502 T^{5} + 571027 T^{6} - 21184370 T^{7} + 139733783 T^{8} + 352011726 T^{9} - 10109995785 T^{10} + 352011726 p T^{11} + 139733783 p^{2} T^{12} - 21184370 p^{3} T^{13} + 571027 p^{4} T^{14} + 129502 p^{5} T^{15} - 9691 p^{6} T^{16} - 1300 p^{7} T^{17} + 317 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 19 T + 200 T^{2} - 1841 T^{3} + 20335 T^{4} - 165842 T^{5} + 1084753 T^{6} - 7135375 T^{7} + 69258390 T^{8} - 291180681 T^{9} + 739941159 T^{10} - 291180681 p T^{11} + 69258390 p^{2} T^{12} - 7135375 p^{3} T^{13} + 1084753 p^{4} T^{14} - 165842 p^{5} T^{15} + 20335 p^{6} T^{16} - 1841 p^{7} T^{17} + 200 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 52 T + 1437 T^{2} + 29168 T^{3} + 487408 T^{4} + 7004616 T^{5} + 89204682 T^{6} + 1029212842 T^{7} + 10906391710 T^{8} + 107272096976 T^{9} + 985695327805 T^{10} + 107272096976 p T^{11} + 10906391710 p^{2} T^{12} + 1029212842 p^{3} T^{13} + 89204682 p^{4} T^{14} + 7004616 p^{5} T^{15} + 487408 p^{6} T^{16} + 29168 p^{7} T^{17} + 1437 p^{8} T^{18} + 52 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 7 T + 186 T^{2} + 1138 T^{3} + 7528 T^{4} + 333473 T^{5} + 1030202 T^{6} + 33499824 T^{7} + 280995592 T^{8} + 1464906960 T^{9} + 39003716995 T^{10} + 1464906960 p T^{11} + 280995592 p^{2} T^{12} + 33499824 p^{3} T^{13} + 1030202 p^{4} T^{14} + 333473 p^{5} T^{15} + 7528 p^{6} T^{16} + 1138 p^{7} T^{17} + 186 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 3 T + 8 T^{2} - 186 T^{3} + 7392 T^{4} - 97351 T^{5} + 1051086 T^{6} - 6302722 T^{7} + 52557350 T^{8} - 658401678 T^{9} + 12985544023 T^{10} - 658401678 p T^{11} + 52557350 p^{2} T^{12} - 6302722 p^{3} T^{13} + 1051086 p^{4} T^{14} - 97351 p^{5} T^{15} + 7392 p^{6} T^{16} - 186 p^{7} T^{17} + 8 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 51 T + 964 T^{2} - 5607 T^{3} - 77007 T^{4} + 1465112 T^{5} - 2994577 T^{6} - 146374009 T^{7} + 1322579600 T^{8} + 8427978077 T^{9} - 215663743901 T^{10} + 8427978077 p T^{11} + 1322579600 p^{2} T^{12} - 146374009 p^{3} T^{13} - 2994577 p^{4} T^{14} + 1465112 p^{5} T^{15} - 77007 p^{6} T^{16} - 5607 p^{7} T^{17} + 964 p^{8} T^{18} - 51 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−5.29925624134385364483494857606, −5.26717863639586492164341322026, −5.14818047363877719105536064508, −5.03237322461664153753809636446, −5.00606364192275380374107413463, −4.88932216475649440393934797356, −4.48902131847679200972959817240, −4.14405422002065102148624426020, −4.08340455478138784358097124592, −3.99709817693010872171911577506, −3.97837959856324366040330949603, −3.83719798466757548392497230868, −3.68011831078985214075761822149, −3.54561354565591521200043278808, −3.14761353504357043299672795388, −2.81424796643085917373179591481, −2.75047125335412103180190272868, −2.27902221926564689561934401534, −2.21667700905905247225843206163, −2.17932492208610412473641262205, −1.89290993201748908837538907324, −1.59472975524456984987053233662, −1.48692967494587366174879011370, −1.34455069556571945032477136924, −1.31641359573412542343833701690, 1.31641359573412542343833701690, 1.34455069556571945032477136924, 1.48692967494587366174879011370, 1.59472975524456984987053233662, 1.89290993201748908837538907324, 2.17932492208610412473641262205, 2.21667700905905247225843206163, 2.27902221926564689561934401534, 2.75047125335412103180190272868, 2.81424796643085917373179591481, 3.14761353504357043299672795388, 3.54561354565591521200043278808, 3.68011831078985214075761822149, 3.83719798466757548392497230868, 3.97837959856324366040330949603, 3.99709817693010872171911577506, 4.08340455478138784358097124592, 4.14405422002065102148624426020, 4.48902131847679200972959817240, 4.88932216475649440393934797356, 5.00606364192275380374107413463, 5.03237322461664153753809636446, 5.14818047363877719105536064508, 5.26717863639586492164341322026, 5.29925624134385364483494857606
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.