Dirichlet series
| L(s) = 1 | + 4·2-s + 3·3-s + 4-s + 16·5-s + 12·6-s + 7-s − 18·8-s − 8·9-s + 64·10-s + 10·11-s + 3·12-s + 7·13-s + 4·14-s + 48·15-s − 29·16-s + 26·17-s − 32·18-s + 16·20-s + 3·21-s + 40·22-s − 13·23-s − 54·24-s + 105·25-s + 28·26-s − 32·27-s + 28-s + 13·29-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.73·3-s + 1/2·4-s + 7.15·5-s + 4.89·6-s + 0.377·7-s − 6.36·8-s − 8/3·9-s + 20.2·10-s + 3.01·11-s + 0.866·12-s + 1.94·13-s + 1.06·14-s + 12.3·15-s − 7.25·16-s + 6.30·17-s − 7.54·18-s + 3.57·20-s + 0.654·21-s + 8.52·22-s − 2.71·23-s − 11.0·24-s + 21·25-s + 5.49·26-s − 6.15·27-s + 0.188·28-s + 2.41·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{13} \cdot 29^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{13} \cdot 29^{13}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(26\) |
| Conductor: | \(23^{13} \cdot 29^{13}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.77484\times 10^{9}\) |
| Root analytic conductor: | \(2.30781\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((26,\ 23^{13} \cdot 29^{13} ,\ ( \ : [1/2]^{13} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(642.3099601\) |
| \(L(\frac12)\) | \(\approx\) | \(642.3099601\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 23 | \( ( 1 + T )^{13} \) |
| 29 | \( ( 1 - T )^{13} \) | |
| good | 2 | \( 1 - p^{2} T + 15 T^{2} - 19 p T^{3} + 47 p T^{4} - 97 p T^{5} + 397 T^{6} - 727 T^{7} + 333 p^{2} T^{8} - 2231 T^{9} + 29 p^{7} T^{10} - 5697 T^{11} + 543 p^{4} T^{12} - 3081 p^{2} T^{13} + 543 p^{5} T^{14} - 5697 p^{2} T^{15} + 29 p^{10} T^{16} - 2231 p^{4} T^{17} + 333 p^{7} T^{18} - 727 p^{6} T^{19} + 397 p^{7} T^{20} - 97 p^{9} T^{21} + 47 p^{10} T^{22} - 19 p^{11} T^{23} + 15 p^{11} T^{24} - p^{14} T^{25} + p^{13} T^{26} \) |
| 3 | \( 1 - p T + 17 T^{2} - 43 T^{3} + 152 T^{4} - 350 T^{5} + 326 p T^{6} - 2059 T^{7} + 1654 p T^{8} - 3197 p T^{9} + 20755 T^{10} - 37030 T^{11} + 301 p^{5} T^{12} - 120584 T^{13} + 301 p^{6} T^{14} - 37030 p^{2} T^{15} + 20755 p^{3} T^{16} - 3197 p^{5} T^{17} + 1654 p^{6} T^{18} - 2059 p^{6} T^{19} + 326 p^{8} T^{20} - 350 p^{8} T^{21} + 152 p^{9} T^{22} - 43 p^{10} T^{23} + 17 p^{11} T^{24} - p^{13} T^{25} + p^{13} T^{26} \) | |
| 5 | \( 1 - 16 T + 151 T^{2} - 1032 T^{3} + 5658 T^{4} - 26139 T^{5} + 105357 T^{6} - 379082 T^{7} + 1239597 T^{8} - 746188 p T^{9} + 10434096 T^{10} - 27277136 T^{11} + 66909704 T^{12} - 154271046 T^{13} + 66909704 p T^{14} - 27277136 p^{2} T^{15} + 10434096 p^{3} T^{16} - 746188 p^{5} T^{17} + 1239597 p^{5} T^{18} - 379082 p^{6} T^{19} + 105357 p^{7} T^{20} - 26139 p^{8} T^{21} + 5658 p^{9} T^{22} - 1032 p^{10} T^{23} + 151 p^{11} T^{24} - 16 p^{12} T^{25} + p^{13} T^{26} \) | |
| 7 | \( 1 - T + 44 T^{2} - 22 T^{3} + 972 T^{4} - 11 T^{5} + 14367 T^{6} + 907 p T^{7} + 23132 p T^{8} + 130615 T^{9} + 1503898 T^{10} + 1533689 T^{11} + 12019984 T^{12} + 1796358 p T^{13} + 12019984 p T^{14} + 1533689 p^{2} T^{15} + 1503898 p^{3} T^{16} + 130615 p^{4} T^{17} + 23132 p^{6} T^{18} + 907 p^{7} T^{19} + 14367 p^{7} T^{20} - 11 p^{8} T^{21} + 972 p^{9} T^{22} - 22 p^{10} T^{23} + 44 p^{11} T^{24} - p^{12} T^{25} + p^{13} T^{26} \) | |
| 11 | \( 1 - 10 T + 129 T^{2} - 87 p T^{3} + 7486 T^{4} - 44779 T^{5} + 267592 T^{6} - 1348334 T^{7} + 6669056 T^{8} - 29027357 T^{9} + 123280921 T^{10} - 470069885 T^{11} + 1745640195 T^{12} - 5867812140 T^{13} + 1745640195 p T^{14} - 470069885 p^{2} T^{15} + 123280921 p^{3} T^{16} - 29027357 p^{4} T^{17} + 6669056 p^{5} T^{18} - 1348334 p^{6} T^{19} + 267592 p^{7} T^{20} - 44779 p^{8} T^{21} + 7486 p^{9} T^{22} - 87 p^{11} T^{23} + 129 p^{11} T^{24} - 10 p^{12} T^{25} + p^{13} T^{26} \) | |
| 13 | \( 1 - 7 T + 133 T^{2} - 776 T^{3} + 8208 T^{4} - 41012 T^{5} + 316653 T^{6} - 1383298 T^{7} + 8656894 T^{8} - 33553565 T^{9} + 179349201 T^{10} - 621906575 T^{11} + 2919955615 T^{12} - 9072659644 T^{13} + 2919955615 p T^{14} - 621906575 p^{2} T^{15} + 179349201 p^{3} T^{16} - 33553565 p^{4} T^{17} + 8656894 p^{5} T^{18} - 1383298 p^{6} T^{19} + 316653 p^{7} T^{20} - 41012 p^{8} T^{21} + 8208 p^{9} T^{22} - 776 p^{10} T^{23} + 133 p^{11} T^{24} - 7 p^{12} T^{25} + p^{13} T^{26} \) | |
| 17 | \( 1 - 26 T + 443 T^{2} - 5624 T^{3} + 59015 T^{4} - 529758 T^{5} + 246635 p T^{6} - 29699383 T^{7} + 190783229 T^{8} - 1119794260 T^{9} + 6045356912 T^{10} - 30129093102 T^{11} + 139068012694 T^{12} - 595198333880 T^{13} + 139068012694 p T^{14} - 30129093102 p^{2} T^{15} + 6045356912 p^{3} T^{16} - 1119794260 p^{4} T^{17} + 190783229 p^{5} T^{18} - 29699383 p^{6} T^{19} + 246635 p^{8} T^{20} - 529758 p^{8} T^{21} + 59015 p^{9} T^{22} - 5624 p^{10} T^{23} + 443 p^{11} T^{24} - 26 p^{12} T^{25} + p^{13} T^{26} \) | |
| 19 | \( 1 + 131 T^{2} - 239 T^{3} + 8327 T^{4} - 28235 T^{5} + 19533 p T^{6} - 1572979 T^{7} + 13335630 T^{8} - 56977156 T^{9} + 391663148 T^{10} - 1539858034 T^{11} + 9262692275 T^{12} - 32779561110 T^{13} + 9262692275 p T^{14} - 1539858034 p^{2} T^{15} + 391663148 p^{3} T^{16} - 56977156 p^{4} T^{17} + 13335630 p^{5} T^{18} - 1572979 p^{6} T^{19} + 19533 p^{8} T^{20} - 28235 p^{8} T^{21} + 8327 p^{9} T^{22} - 239 p^{10} T^{23} + 131 p^{11} T^{24} + p^{13} T^{26} \) | |
| 31 | \( 1 + 6 T + 219 T^{2} + 1177 T^{3} + 23675 T^{4} + 114348 T^{5} + 1673197 T^{6} + 7240633 T^{7} + 87128432 T^{8} + 338263472 T^{9} + 3614888285 T^{10} + 12755260394 T^{11} + 127117776313 T^{12} + 417702954404 T^{13} + 127117776313 p T^{14} + 12755260394 p^{2} T^{15} + 3614888285 p^{3} T^{16} + 338263472 p^{4} T^{17} + 87128432 p^{5} T^{18} + 7240633 p^{6} T^{19} + 1673197 p^{7} T^{20} + 114348 p^{8} T^{21} + 23675 p^{9} T^{22} + 1177 p^{10} T^{23} + 219 p^{11} T^{24} + 6 p^{12} T^{25} + p^{13} T^{26} \) | |
| 37 | \( 1 - 15 T + 404 T^{2} - 4658 T^{3} + 73032 T^{4} - 695636 T^{5} + 8157306 T^{6} - 66709445 T^{7} + 642020904 T^{8} - 4610206838 T^{9} + 1030217863 p T^{10} - 243399565516 T^{11} + 1771690302159 T^{12} - 10108357803174 T^{13} + 1771690302159 p T^{14} - 243399565516 p^{2} T^{15} + 1030217863 p^{4} T^{16} - 4610206838 p^{4} T^{17} + 642020904 p^{5} T^{18} - 66709445 p^{6} T^{19} + 8157306 p^{7} T^{20} - 695636 p^{8} T^{21} + 73032 p^{9} T^{22} - 4658 p^{10} T^{23} + 404 p^{11} T^{24} - 15 p^{12} T^{25} + p^{13} T^{26} \) | |
| 41 | \( 1 - 9 T + 312 T^{2} - 2384 T^{3} + 1129 p T^{4} - 307268 T^{5} + 4430238 T^{6} - 25887442 T^{7} + 312053432 T^{8} - 1627381936 T^{9} + 17524169596 T^{10} - 83059990388 T^{11} + 828356088124 T^{12} - 3638235256906 T^{13} + 828356088124 p T^{14} - 83059990388 p^{2} T^{15} + 17524169596 p^{3} T^{16} - 1627381936 p^{4} T^{17} + 312053432 p^{5} T^{18} - 25887442 p^{6} T^{19} + 4430238 p^{7} T^{20} - 307268 p^{8} T^{21} + 1129 p^{10} T^{22} - 2384 p^{10} T^{23} + 312 p^{11} T^{24} - 9 p^{12} T^{25} + p^{13} T^{26} \) | |
| 43 | \( 1 - T + 309 T^{2} - 5 p T^{3} + 44138 T^{4} - 26154 T^{5} + 2091 p^{2} T^{6} - 3087745 T^{7} + 235173405 T^{8} - 334018968 T^{9} + 11008331369 T^{10} - 26221003985 T^{11} + 457110290538 T^{12} - 1382586516876 T^{13} + 457110290538 p T^{14} - 26221003985 p^{2} T^{15} + 11008331369 p^{3} T^{16} - 334018968 p^{4} T^{17} + 235173405 p^{5} T^{18} - 3087745 p^{6} T^{19} + 2091 p^{9} T^{20} - 26154 p^{8} T^{21} + 44138 p^{9} T^{22} - 5 p^{11} T^{23} + 309 p^{11} T^{24} - p^{12} T^{25} + p^{13} T^{26} \) | |
| 47 | \( 1 - 15 T + 538 T^{2} - 6818 T^{3} + 134254 T^{4} - 1473084 T^{5} + 20819660 T^{6} - 200522045 T^{7} + 2252696337 T^{8} - 19173221203 T^{9} + 180161912833 T^{10} - 1357141814693 T^{11} + 10973804183039 T^{12} - 72929252644700 T^{13} + 10973804183039 p T^{14} - 1357141814693 p^{2} T^{15} + 180161912833 p^{3} T^{16} - 19173221203 p^{4} T^{17} + 2252696337 p^{5} T^{18} - 200522045 p^{6} T^{19} + 20819660 p^{7} T^{20} - 1473084 p^{8} T^{21} + 134254 p^{9} T^{22} - 6818 p^{10} T^{23} + 538 p^{11} T^{24} - 15 p^{12} T^{25} + p^{13} T^{26} \) | |
| 53 | \( 1 - 43 T + 1315 T^{2} - 28796 T^{3} + 527234 T^{4} - 152817 p T^{5} + 110085139 T^{6} - 1326292478 T^{7} + 14556860056 T^{8} - 145603343341 T^{9} + 1350391398871 T^{10} - 11596807768092 T^{11} + 93303826685684 T^{12} - 700534218536522 T^{13} + 93303826685684 p T^{14} - 11596807768092 p^{2} T^{15} + 1350391398871 p^{3} T^{16} - 145603343341 p^{4} T^{17} + 14556860056 p^{5} T^{18} - 1326292478 p^{6} T^{19} + 110085139 p^{7} T^{20} - 152817 p^{9} T^{21} + 527234 p^{9} T^{22} - 28796 p^{10} T^{23} + 1315 p^{11} T^{24} - 43 p^{12} T^{25} + p^{13} T^{26} \) | |
| 59 | \( 1 + 9 T + 248 T^{2} + 3047 T^{3} + 48131 T^{4} + 494885 T^{5} + 6608818 T^{6} + 62816390 T^{7} + 687722697 T^{8} + 6044502246 T^{9} + 59622236678 T^{10} + 470861754348 T^{11} + 4184281778788 T^{12} + 30794504149682 T^{13} + 4184281778788 p T^{14} + 470861754348 p^{2} T^{15} + 59622236678 p^{3} T^{16} + 6044502246 p^{4} T^{17} + 687722697 p^{5} T^{18} + 62816390 p^{6} T^{19} + 6608818 p^{7} T^{20} + 494885 p^{8} T^{21} + 48131 p^{9} T^{22} + 3047 p^{10} T^{23} + 248 p^{11} T^{24} + 9 p^{12} T^{25} + p^{13} T^{26} \) | |
| 61 | \( 1 - 20 T + 491 T^{2} - 6662 T^{3} + 102320 T^{4} - 1158288 T^{5} + 14386931 T^{6} - 146172204 T^{7} + 1564997432 T^{8} - 14416278202 T^{9} + 136902142177 T^{10} - 1156374492284 T^{11} + 9961381576960 T^{12} - 77273889474880 T^{13} + 9961381576960 p T^{14} - 1156374492284 p^{2} T^{15} + 136902142177 p^{3} T^{16} - 14416278202 p^{4} T^{17} + 1564997432 p^{5} T^{18} - 146172204 p^{6} T^{19} + 14386931 p^{7} T^{20} - 1158288 p^{8} T^{21} + 102320 p^{9} T^{22} - 6662 p^{10} T^{23} + 491 p^{11} T^{24} - 20 p^{12} T^{25} + p^{13} T^{26} \) | |
| 67 | \( 1 - T + 345 T^{2} - 1033 T^{3} + 60225 T^{4} - 315162 T^{5} + 7588143 T^{6} - 51308407 T^{7} + 804912719 T^{8} - 5664080339 T^{9} + 74276768870 T^{10} - 488828629460 T^{11} + 5871020311353 T^{12} - 35427149740492 T^{13} + 5871020311353 p T^{14} - 488828629460 p^{2} T^{15} + 74276768870 p^{3} T^{16} - 5664080339 p^{4} T^{17} + 804912719 p^{5} T^{18} - 51308407 p^{6} T^{19} + 7588143 p^{7} T^{20} - 315162 p^{8} T^{21} + 60225 p^{9} T^{22} - 1033 p^{10} T^{23} + 345 p^{11} T^{24} - p^{12} T^{25} + p^{13} T^{26} \) | |
| 71 | \( 1 - 17 T + 411 T^{2} - 4819 T^{3} + 72983 T^{4} - 696550 T^{5} + 8552058 T^{6} - 72895639 T^{7} + 826016480 T^{8} - 6823981862 T^{9} + 75438996290 T^{10} - 609685710595 T^{11} + 6338553518754 T^{12} - 47648490220692 T^{13} + 6338553518754 p T^{14} - 609685710595 p^{2} T^{15} + 75438996290 p^{3} T^{16} - 6823981862 p^{4} T^{17} + 826016480 p^{5} T^{18} - 72895639 p^{6} T^{19} + 8552058 p^{7} T^{20} - 696550 p^{8} T^{21} + 72983 p^{9} T^{22} - 4819 p^{10} T^{23} + 411 p^{11} T^{24} - 17 p^{12} T^{25} + p^{13} T^{26} \) | |
| 73 | \( 1 - 26 T + 771 T^{2} - 14457 T^{3} + 266822 T^{4} - 3998594 T^{5} + 57241185 T^{6} - 724874106 T^{7} + 8706447990 T^{8} - 96138348524 T^{9} + 1005759688137 T^{10} - 9859920644507 T^{11} + 91592280066474 T^{12} - 803840006556492 T^{13} + 91592280066474 p T^{14} - 9859920644507 p^{2} T^{15} + 1005759688137 p^{3} T^{16} - 96138348524 p^{4} T^{17} + 8706447990 p^{5} T^{18} - 724874106 p^{6} T^{19} + 57241185 p^{7} T^{20} - 3998594 p^{8} T^{21} + 266822 p^{9} T^{22} - 14457 p^{10} T^{23} + 771 p^{11} T^{24} - 26 p^{12} T^{25} + p^{13} T^{26} \) | |
| 79 | \( 1 - 5 T + 339 T^{2} - 2040 T^{3} + 58251 T^{4} - 365198 T^{5} + 7585540 T^{6} - 45405015 T^{7} + 871129057 T^{8} - 4953738105 T^{9} + 88251798724 T^{10} - 484381708717 T^{11} + 7805005889985 T^{12} - 41126444625392 T^{13} + 7805005889985 p T^{14} - 484381708717 p^{2} T^{15} + 88251798724 p^{3} T^{16} - 4953738105 p^{4} T^{17} + 871129057 p^{5} T^{18} - 45405015 p^{6} T^{19} + 7585540 p^{7} T^{20} - 365198 p^{8} T^{21} + 58251 p^{9} T^{22} - 2040 p^{10} T^{23} + 339 p^{11} T^{24} - 5 p^{12} T^{25} + p^{13} T^{26} \) | |
| 83 | \( 1 - 4 T + 589 T^{2} - 2547 T^{3} + 182867 T^{4} - 825229 T^{5} + 38696874 T^{6} - 176614203 T^{7} + 6158379204 T^{8} - 27559881015 T^{9} + 773307881825 T^{10} - 3290559088410 T^{11} + 78527868396960 T^{12} - 307578321495216 T^{13} + 78527868396960 p T^{14} - 3290559088410 p^{2} T^{15} + 773307881825 p^{3} T^{16} - 27559881015 p^{4} T^{17} + 6158379204 p^{5} T^{18} - 176614203 p^{6} T^{19} + 38696874 p^{7} T^{20} - 825229 p^{8} T^{21} + 182867 p^{9} T^{22} - 2547 p^{10} T^{23} + 589 p^{11} T^{24} - 4 p^{12} T^{25} + p^{13} T^{26} \) | |
| 89 | \( 1 - 48 T + 1594 T^{2} - 38770 T^{3} + 792883 T^{4} - 13825325 T^{5} + 215573747 T^{6} - 3023761998 T^{7} + 39145660911 T^{8} - 469173717361 T^{9} + 5295811844460 T^{10} - 56323668994780 T^{11} + 571147707266589 T^{12} - 5506317450879454 T^{13} + 571147707266589 p T^{14} - 56323668994780 p^{2} T^{15} + 5295811844460 p^{3} T^{16} - 469173717361 p^{4} T^{17} + 39145660911 p^{5} T^{18} - 3023761998 p^{6} T^{19} + 215573747 p^{7} T^{20} - 13825325 p^{8} T^{21} + 792883 p^{9} T^{22} - 38770 p^{10} T^{23} + 1594 p^{11} T^{24} - 48 p^{12} T^{25} + p^{13} T^{26} \) | |
| 97 | \( 1 - 30 T + 1113 T^{2} - 22870 T^{3} + 505329 T^{4} - 8025669 T^{5} + 133001593 T^{6} - 1736062668 T^{7} + 23652981736 T^{8} - 265217381024 T^{9} + 3155425042003 T^{10} - 31742477150988 T^{11} + 346837094966997 T^{12} - 3257920191910950 T^{13} + 346837094966997 p T^{14} - 31742477150988 p^{2} T^{15} + 3155425042003 p^{3} T^{16} - 265217381024 p^{4} T^{17} + 23652981736 p^{5} T^{18} - 1736062668 p^{6} T^{19} + 133001593 p^{7} T^{20} - 8025669 p^{8} T^{21} + 505329 p^{9} T^{22} - 22870 p^{10} T^{23} + 1113 p^{11} T^{24} - 30 p^{12} T^{25} + p^{13} T^{26} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{26} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.34387394586028050846093217469, −3.15603630959753957330193980861, −3.08921814517141910017729802912, −2.85949544639081381180237073440, −2.85582297696186096288780121729, −2.79440050873709987585568695825, −2.77171177054327030967030352099, −2.58099701622677513943113391642, −2.47405650182115465318521138406, −2.34553721257791121360180092001, −2.30768569437207625735976389202, −2.23231149894108566865519364338, −1.99030016561006839685257089590, −1.91658444091893646947894067103, −1.88522951772703649404046500246, −1.86981339596634775435412588347, −1.79591863825321990167706243192, −1.53106694196955844791323278077, −1.33612108576794515376688237978, −1.30500374845837214906970266553, −1.06157068357453934421540025652, −0.903343790852595231006206140851, −0.72228352570196251496121409999, −0.65207629588088117775402841103, −0.64399188340823565624343909787, 0.64399188340823565624343909787, 0.65207629588088117775402841103, 0.72228352570196251496121409999, 0.903343790852595231006206140851, 1.06157068357453934421540025652, 1.30500374845837214906970266553, 1.33612108576794515376688237978, 1.53106694196955844791323278077, 1.79591863825321990167706243192, 1.86981339596634775435412588347, 1.88522951772703649404046500246, 1.91658444091893646947894067103, 1.99030016561006839685257089590, 2.23231149894108566865519364338, 2.30768569437207625735976389202, 2.34553721257791121360180092001, 2.47405650182115465318521138406, 2.58099701622677513943113391642, 2.77171177054327030967030352099, 2.79440050873709987585568695825, 2.85582297696186096288780121729, 2.85949544639081381180237073440, 3.08921814517141910017729802912, 3.15603630959753957330193980861, 3.34387394586028050846093217469
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.