Dirichlet series
| L(s) = 1 | − 4·3-s − 6·5-s + 2·7-s − 9-s − 10·11-s − 4·13-s + 24·15-s + 10·17-s − 8·19-s − 8·21-s − 18·23-s − 3·25-s + 24·27-s − 14·29-s − 4·31-s + 40·33-s − 12·35-s − 16·37-s + 16·39-s + 10·41-s − 12·43-s + 6·45-s + 13·47-s − 39·49-s − 40·51-s − 26·53-s + 60·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 2.68·5-s + 0.755·7-s − 1/3·9-s − 3.01·11-s − 1.10·13-s + 6.19·15-s + 2.42·17-s − 1.83·19-s − 1.74·21-s − 3.75·23-s − 3/5·25-s + 4.61·27-s − 2.59·29-s − 0.718·31-s + 6.96·33-s − 2.02·35-s − 2.63·37-s + 2.56·39-s + 1.56·41-s − 1.82·43-s + 0.894·45-s + 1.89·47-s − 5.57·49-s − 5.60·51-s − 3.57·53-s + 8.09·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{91} \cdot 47^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{91} \cdot 47^{13}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{13} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(26\) |
| Conductor: | \(2^{91} \cdot 47^{13}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(7.25444\times 10^{21}\) |
| Root analytic conductor: | \(6.93094\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(13\) |
| Selberg data: | \((26,\ 2^{91} \cdot 47^{13} ,\ ( \ : [1/2]^{13} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 47 | \( ( 1 - T )^{13} \) | |
| good | 3 | \( 1 + 4 T + 17 T^{2} + 16 p T^{3} + 139 T^{4} + 112 p T^{5} + 265 p T^{6} + 1682 T^{7} + 3524 T^{8} + 6806 T^{9} + 4382 p T^{10} + 23654 T^{11} + 4798 p^{2} T^{12} + 74140 T^{13} + 4798 p^{3} T^{14} + 23654 p^{2} T^{15} + 4382 p^{4} T^{16} + 6806 p^{4} T^{17} + 3524 p^{5} T^{18} + 1682 p^{6} T^{19} + 265 p^{8} T^{20} + 112 p^{9} T^{21} + 139 p^{9} T^{22} + 16 p^{11} T^{23} + 17 p^{11} T^{24} + 4 p^{12} T^{25} + p^{13} T^{26} \) |
| 5 | \( 1 + 6 T + 39 T^{2} + 34 p T^{3} + 718 T^{4} + 2584 T^{5} + 8834 T^{6} + 27538 T^{7} + 81729 T^{8} + 226546 T^{9} + 602183 T^{10} + 1506988 T^{11} + 3635352 T^{12} + 8270864 T^{13} + 3635352 p T^{14} + 1506988 p^{2} T^{15} + 602183 p^{3} T^{16} + 226546 p^{4} T^{17} + 81729 p^{5} T^{18} + 27538 p^{6} T^{19} + 8834 p^{7} T^{20} + 2584 p^{8} T^{21} + 718 p^{9} T^{22} + 34 p^{11} T^{23} + 39 p^{11} T^{24} + 6 p^{12} T^{25} + p^{13} T^{26} \) | |
| 7 | \( 1 - 2 T + 43 T^{2} - 74 T^{3} + 941 T^{4} - 1356 T^{5} + 2007 p T^{6} - 16748 T^{7} + 161544 T^{8} - 159150 T^{9} + 1523588 T^{10} - 1270786 T^{11} + 12247846 T^{12} - 9186328 T^{13} + 12247846 p T^{14} - 1270786 p^{2} T^{15} + 1523588 p^{3} T^{16} - 159150 p^{4} T^{17} + 161544 p^{5} T^{18} - 16748 p^{6} T^{19} + 2007 p^{8} T^{20} - 1356 p^{8} T^{21} + 941 p^{9} T^{22} - 74 p^{10} T^{23} + 43 p^{11} T^{24} - 2 p^{12} T^{25} + p^{13} T^{26} \) | |
| 11 | \( 1 + 10 T + 9 p T^{2} + 650 T^{3} + 4058 T^{4} + 20652 T^{5} + 99178 T^{6} + 414690 T^{7} + 1652187 T^{8} + 5930846 T^{9} + 20771961 T^{10} + 67868140 T^{11} + 226158528 T^{12} + 731794280 T^{13} + 226158528 p T^{14} + 67868140 p^{2} T^{15} + 20771961 p^{3} T^{16} + 5930846 p^{4} T^{17} + 1652187 p^{5} T^{18} + 414690 p^{6} T^{19} + 99178 p^{7} T^{20} + 20652 p^{8} T^{21} + 4058 p^{9} T^{22} + 650 p^{10} T^{23} + 9 p^{12} T^{24} + 10 p^{12} T^{25} + p^{13} T^{26} \) | |
| 13 | \( 1 + 4 T + 57 T^{2} + 162 T^{3} + 1706 T^{4} + 4756 T^{5} + 40298 T^{6} + 107526 T^{7} + 758747 T^{8} + 1984020 T^{9} + 12573747 T^{10} + 31907560 T^{11} + 181279940 T^{12} + 434738504 T^{13} + 181279940 p T^{14} + 31907560 p^{2} T^{15} + 12573747 p^{3} T^{16} + 1984020 p^{4} T^{17} + 758747 p^{5} T^{18} + 107526 p^{6} T^{19} + 40298 p^{7} T^{20} + 4756 p^{8} T^{21} + 1706 p^{9} T^{22} + 162 p^{10} T^{23} + 57 p^{11} T^{24} + 4 p^{12} T^{25} + p^{13} T^{26} \) | |
| 17 | \( 1 - 10 T + 159 T^{2} - 1144 T^{3} + 10655 T^{4} - 60322 T^{5} + 1461 p^{2} T^{6} - 1960636 T^{7} + 671148 p T^{8} - 44670952 T^{9} + 233071850 T^{10} - 805433516 T^{11} + 4082487178 T^{12} - 13519828736 T^{13} + 4082487178 p T^{14} - 805433516 p^{2} T^{15} + 233071850 p^{3} T^{16} - 44670952 p^{4} T^{17} + 671148 p^{6} T^{18} - 1960636 p^{6} T^{19} + 1461 p^{9} T^{20} - 60322 p^{8} T^{21} + 10655 p^{9} T^{22} - 1144 p^{10} T^{23} + 159 p^{11} T^{24} - 10 p^{12} T^{25} + p^{13} T^{26} \) | |
| 19 | \( 1 + 8 T + 101 T^{2} + 542 T^{3} + 4080 T^{4} + 16068 T^{5} + 98348 T^{6} + 15558 p T^{7} + 1949009 T^{8} + 5458848 T^{9} + 43927245 T^{10} + 135331040 T^{11} + 1016846988 T^{12} + 3006254856 T^{13} + 1016846988 p T^{14} + 135331040 p^{2} T^{15} + 43927245 p^{3} T^{16} + 5458848 p^{4} T^{17} + 1949009 p^{5} T^{18} + 15558 p^{7} T^{19} + 98348 p^{7} T^{20} + 16068 p^{8} T^{21} + 4080 p^{9} T^{22} + 542 p^{10} T^{23} + 101 p^{11} T^{24} + 8 p^{12} T^{25} + p^{13} T^{26} \) | |
| 23 | \( 1 + 18 T + 305 T^{2} + 3514 T^{3} + 37136 T^{4} + 325668 T^{5} + 2646392 T^{6} + 19051934 T^{7} + 128728281 T^{8} + 794783542 T^{9} + 4658817553 T^{10} + 25389597448 T^{11} + 132491485696 T^{12} + 648297296808 T^{13} + 132491485696 p T^{14} + 25389597448 p^{2} T^{15} + 4658817553 p^{3} T^{16} + 794783542 p^{4} T^{17} + 128728281 p^{5} T^{18} + 19051934 p^{6} T^{19} + 2646392 p^{7} T^{20} + 325668 p^{8} T^{21} + 37136 p^{9} T^{22} + 3514 p^{10} T^{23} + 305 p^{11} T^{24} + 18 p^{12} T^{25} + p^{13} T^{26} \) | |
| 29 | \( 1 + 14 T + 333 T^{2} + 3734 T^{3} + 51010 T^{4} + 475156 T^{5} + 4815362 T^{6} + 38263902 T^{7} + 315851919 T^{8} + 2180821162 T^{9} + 528845239 p T^{10} + 93085872148 T^{11} + 570906321012 T^{12} + 3061660588728 T^{13} + 570906321012 p T^{14} + 93085872148 p^{2} T^{15} + 528845239 p^{4} T^{16} + 2180821162 p^{4} T^{17} + 315851919 p^{5} T^{18} + 38263902 p^{6} T^{19} + 4815362 p^{7} T^{20} + 475156 p^{8} T^{21} + 51010 p^{9} T^{22} + 3734 p^{10} T^{23} + 333 p^{11} T^{24} + 14 p^{12} T^{25} + p^{13} T^{26} \) | |
| 31 | \( 1 + 4 T + 239 T^{2} + 906 T^{3} + 28368 T^{4} + 99112 T^{5} + 2237064 T^{6} + 7142474 T^{7} + 131575251 T^{8} + 384647452 T^{9} + 6112924669 T^{10} + 16368870892 T^{11} + 230736218000 T^{12} + 562852408048 T^{13} + 230736218000 p T^{14} + 16368870892 p^{2} T^{15} + 6112924669 p^{3} T^{16} + 384647452 p^{4} T^{17} + 131575251 p^{5} T^{18} + 7142474 p^{6} T^{19} + 2237064 p^{7} T^{20} + 99112 p^{8} T^{21} + 28368 p^{9} T^{22} + 906 p^{10} T^{23} + 239 p^{11} T^{24} + 4 p^{12} T^{25} + p^{13} T^{26} \) | |
| 37 | \( 1 + 16 T + 387 T^{2} + 4842 T^{3} + 69421 T^{4} + 723480 T^{5} + 7862893 T^{6} + 70420010 T^{7} + 633309636 T^{8} + 134039850 p T^{9} + 38427836622 T^{10} + 265750638756 T^{11} + 1809250029880 T^{12} + 299947921260 p T^{13} + 1809250029880 p T^{14} + 265750638756 p^{2} T^{15} + 38427836622 p^{3} T^{16} + 134039850 p^{5} T^{17} + 633309636 p^{5} T^{18} + 70420010 p^{6} T^{19} + 7862893 p^{7} T^{20} + 723480 p^{8} T^{21} + 69421 p^{9} T^{22} + 4842 p^{10} T^{23} + 387 p^{11} T^{24} + 16 p^{12} T^{25} + p^{13} T^{26} \) | |
| 41 | \( 1 - 10 T + 283 T^{2} - 2126 T^{3} + 35838 T^{4} - 195204 T^{5} + 2615990 T^{6} - 5090 p^{2} T^{7} + 115921277 T^{8} + 2704026 T^{9} + 2770228559 T^{10} + 23541984656 T^{11} + 16764701796 T^{12} + 1447866430888 T^{13} + 16764701796 p T^{14} + 23541984656 p^{2} T^{15} + 2770228559 p^{3} T^{16} + 2704026 p^{4} T^{17} + 115921277 p^{5} T^{18} - 5090 p^{8} T^{19} + 2615990 p^{7} T^{20} - 195204 p^{8} T^{21} + 35838 p^{9} T^{22} - 2126 p^{10} T^{23} + 283 p^{11} T^{24} - 10 p^{12} T^{25} + p^{13} T^{26} \) | |
| 43 | \( 1 + 12 T + 339 T^{2} + 3606 T^{3} + 55398 T^{4} + 530800 T^{5} + 137186 p T^{6} + 51617958 T^{7} + 465699971 T^{8} + 3758156420 T^{9} + 29192394481 T^{10} + 217849810828 T^{11} + 1504990594496 T^{12} + 10324539522064 T^{13} + 1504990594496 p T^{14} + 217849810828 p^{2} T^{15} + 29192394481 p^{3} T^{16} + 3758156420 p^{4} T^{17} + 465699971 p^{5} T^{18} + 51617958 p^{6} T^{19} + 137186 p^{8} T^{20} + 530800 p^{8} T^{21} + 55398 p^{9} T^{22} + 3606 p^{10} T^{23} + 339 p^{11} T^{24} + 12 p^{12} T^{25} + p^{13} T^{26} \) | |
| 53 | \( 1 + 26 T + 685 T^{2} + 11638 T^{3} + 192613 T^{4} + 2550640 T^{5} + 32708011 T^{6} + 361005744 T^{7} + 3860065380 T^{8} + 36727550772 T^{9} + 339012997604 T^{10} + 2830531783778 T^{11} + 22951742789562 T^{12} + 169519617790676 T^{13} + 22951742789562 p T^{14} + 2830531783778 p^{2} T^{15} + 339012997604 p^{3} T^{16} + 36727550772 p^{4} T^{17} + 3860065380 p^{5} T^{18} + 361005744 p^{6} T^{19} + 32708011 p^{7} T^{20} + 2550640 p^{8} T^{21} + 192613 p^{9} T^{22} + 11638 p^{10} T^{23} + 685 p^{11} T^{24} + 26 p^{12} T^{25} + p^{13} T^{26} \) | |
| 59 | \( 1 + 30 T + 913 T^{2} + 17338 T^{3} + 309975 T^{4} + 4354138 T^{5} + 57090923 T^{6} + 638072946 T^{7} + 6691277736 T^{8} + 62616621148 T^{9} + 558068596550 T^{10} + 4615099515516 T^{11} + 37220300810970 T^{12} + 287476961586248 T^{13} + 37220300810970 p T^{14} + 4615099515516 p^{2} T^{15} + 558068596550 p^{3} T^{16} + 62616621148 p^{4} T^{17} + 6691277736 p^{5} T^{18} + 638072946 p^{6} T^{19} + 57090923 p^{7} T^{20} + 4354138 p^{8} T^{21} + 309975 p^{9} T^{22} + 17338 p^{10} T^{23} + 913 p^{11} T^{24} + 30 p^{12} T^{25} + p^{13} T^{26} \) | |
| 61 | \( 1 + 18 T + 649 T^{2} + 9986 T^{3} + 201257 T^{4} + 2667564 T^{5} + 39440763 T^{6} + 453983044 T^{7} + 5448608768 T^{8} + 54852935628 T^{9} + 9183873852 p T^{10} + 4958943015746 T^{11} + 44095318019126 T^{12} + 344010714552380 T^{13} + 44095318019126 p T^{14} + 4958943015746 p^{2} T^{15} + 9183873852 p^{4} T^{16} + 54852935628 p^{4} T^{17} + 5448608768 p^{5} T^{18} + 453983044 p^{6} T^{19} + 39440763 p^{7} T^{20} + 2667564 p^{8} T^{21} + 201257 p^{9} T^{22} + 9986 p^{10} T^{23} + 649 p^{11} T^{24} + 18 p^{12} T^{25} + p^{13} T^{26} \) | |
| 67 | \( 1 + 4 T + 495 T^{2} + 954 T^{3} + 111592 T^{4} + 9264 T^{5} + 15716876 T^{6} - 25271854 T^{7} + 1605262459 T^{8} - 4976176156 T^{9} + 132022621149 T^{10} - 552382727860 T^{11} + 9534631107900 T^{12} - 43065297690560 T^{13} + 9534631107900 p T^{14} - 552382727860 p^{2} T^{15} + 132022621149 p^{3} T^{16} - 4976176156 p^{4} T^{17} + 1605262459 p^{5} T^{18} - 25271854 p^{6} T^{19} + 15716876 p^{7} T^{20} + 9264 p^{8} T^{21} + 111592 p^{9} T^{22} + 954 p^{10} T^{23} + 495 p^{11} T^{24} + 4 p^{12} T^{25} + p^{13} T^{26} \) | |
| 71 | \( 1 + 36 T + 1137 T^{2} + 24680 T^{3} + 483457 T^{4} + 7852114 T^{5} + 117670391 T^{6} + 1555822768 T^{7} + 19259706764 T^{8} + 216508036136 T^{9} + 2300410602674 T^{10} + 22528250174488 T^{11} + 209677334974696 T^{12} + 1810640137650212 T^{13} + 209677334974696 p T^{14} + 22528250174488 p^{2} T^{15} + 2300410602674 p^{3} T^{16} + 216508036136 p^{4} T^{17} + 19259706764 p^{5} T^{18} + 1555822768 p^{6} T^{19} + 117670391 p^{7} T^{20} + 7852114 p^{8} T^{21} + 483457 p^{9} T^{22} + 24680 p^{10} T^{23} + 1137 p^{11} T^{24} + 36 p^{12} T^{25} + p^{13} T^{26} \) | |
| 73 | \( 1 - 10 T + 421 T^{2} - 2978 T^{3} + 77486 T^{4} - 339084 T^{5} + 8288554 T^{6} - 10520782 T^{7} + 609481099 T^{8} + 1640739850 T^{9} + 39305346439 T^{10} + 258779684912 T^{11} + 2764318667232 T^{12} + 22000169901656 T^{13} + 2764318667232 p T^{14} + 258779684912 p^{2} T^{15} + 39305346439 p^{3} T^{16} + 1640739850 p^{4} T^{17} + 609481099 p^{5} T^{18} - 10520782 p^{6} T^{19} + 8288554 p^{7} T^{20} - 339084 p^{8} T^{21} + 77486 p^{9} T^{22} - 2978 p^{10} T^{23} + 421 p^{11} T^{24} - 10 p^{12} T^{25} + p^{13} T^{26} \) | |
| 79 | \( 1 + 149 T^{2} + 564 T^{3} + 26745 T^{4} + 110714 T^{5} + 3945667 T^{6} + 13776944 T^{7} + 438680120 T^{8} + 1893670824 T^{9} + 46008636418 T^{10} + 189544266748 T^{11} + 4101925241336 T^{12} + 14824119105532 T^{13} + 4101925241336 p T^{14} + 189544266748 p^{2} T^{15} + 46008636418 p^{3} T^{16} + 1893670824 p^{4} T^{17} + 438680120 p^{5} T^{18} + 13776944 p^{6} T^{19} + 3945667 p^{7} T^{20} + 110714 p^{8} T^{21} + 26745 p^{9} T^{22} + 564 p^{10} T^{23} + 149 p^{11} T^{24} + p^{13} T^{26} \) | |
| 83 | \( 1 + 12 T + 319 T^{2} + 4520 T^{3} + 72838 T^{4} + 11648 p T^{5} + 12561522 T^{6} + 149129032 T^{7} + 1730470995 T^{8} + 18449656564 T^{9} + 196333746981 T^{10} + 1931733794768 T^{11} + 18847316508148 T^{12} + 172459787424384 T^{13} + 18847316508148 p T^{14} + 1931733794768 p^{2} T^{15} + 196333746981 p^{3} T^{16} + 18449656564 p^{4} T^{17} + 1730470995 p^{5} T^{18} + 149129032 p^{6} T^{19} + 12561522 p^{7} T^{20} + 11648 p^{9} T^{21} + 72838 p^{9} T^{22} + 4520 p^{10} T^{23} + 319 p^{11} T^{24} + 12 p^{12} T^{25} + p^{13} T^{26} \) | |
| 89 | \( 1 - 50 T + 1587 T^{2} - 36660 T^{3} + 698583 T^{4} - 11246054 T^{5} + 158804785 T^{6} - 1979141704 T^{7} + 22241935380 T^{8} - 226371590856 T^{9} + 2138323619974 T^{10} - 19097674945796 T^{11} + 169672090960558 T^{12} - 1552349392519880 T^{13} + 169672090960558 p T^{14} - 19097674945796 p^{2} T^{15} + 2138323619974 p^{3} T^{16} - 226371590856 p^{4} T^{17} + 22241935380 p^{5} T^{18} - 1979141704 p^{6} T^{19} + 158804785 p^{7} T^{20} - 11246054 p^{8} T^{21} + 698583 p^{9} T^{22} - 36660 p^{10} T^{23} + 1587 p^{11} T^{24} - 50 p^{12} T^{25} + p^{13} T^{26} \) | |
| 97 | \( 1 + 10 T + 707 T^{2} + 7892 T^{3} + 262079 T^{4} + 2920210 T^{5} + 66548437 T^{6} + 700888200 T^{7} + 12650340544 T^{8} + 123586793152 T^{9} + 1883565030914 T^{10} + 16883461405644 T^{11} + 225253830634038 T^{12} + 1830944139026712 T^{13} + 225253830634038 p T^{14} + 16883461405644 p^{2} T^{15} + 1883565030914 p^{3} T^{16} + 123586793152 p^{4} T^{17} + 12650340544 p^{5} T^{18} + 700888200 p^{6} T^{19} + 66548437 p^{7} T^{20} + 2920210 p^{8} T^{21} + 262079 p^{9} T^{22} + 7892 p^{10} T^{23} + 707 p^{11} T^{24} + 10 p^{12} T^{25} + p^{13} T^{26} \) | |
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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
−2.67088669567428546987872309625, −2.51081677889604202653238566272, −2.40233077569458728183592326375, −2.36298860469408128770109077778, −2.35314071535041607986409681864, −2.34963514802327499192805813209, −2.33294364609376638361892346727, −2.28527925744341170027384999465, −2.25429310884927247805565025187, −2.09046183735231231338572945254, −2.02505993864399053528627264903, −1.78898078346853243884928454084, −1.78381796121970188443468659779, −1.73043597984398659567615677362, −1.71949972559482819121075616065, −1.49586946256358256881131868669, −1.44547678808662664207859881148, −1.27736453775548621928863023221, −1.25196347213267557681546822190, −1.23969936329321802645361790705, −1.23783860410821726942744589382, −1.20967211884178545716979535483, −1.10193251011809689454789831041, −0.813659102365090373874298193587, −0.78595787786768713221782722819, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.78595787786768713221782722819, 0.813659102365090373874298193587, 1.10193251011809689454789831041, 1.20967211884178545716979535483, 1.23783860410821726942744589382, 1.23969936329321802645361790705, 1.25196347213267557681546822190, 1.27736453775548621928863023221, 1.44547678808662664207859881148, 1.49586946256358256881131868669, 1.71949972559482819121075616065, 1.73043597984398659567615677362, 1.78381796121970188443468659779, 1.78898078346853243884928454084, 2.02505993864399053528627264903, 2.09046183735231231338572945254, 2.25429310884927247805565025187, 2.28527925744341170027384999465, 2.33294364609376638361892346727, 2.34963514802327499192805813209, 2.35314071535041607986409681864, 2.36298860469408128770109077778, 2.40233077569458728183592326375, 2.51081677889604202653238566272, 2.67088669567428546987872309625
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.