Dirichlet series
| L(s) = 1 | + 4·2-s − 13·3-s + 12·5-s − 52·6-s + 13·7-s − 21·8-s + 91·9-s + 48·10-s + 3·11-s + 21·13-s + 52·14-s − 156·15-s − 23·16-s + 17·17-s + 364·18-s + 5·19-s − 169·21-s + 12·22-s + 4·23-s + 273·24-s + 40·25-s + 84·26-s − 455·27-s + 21·29-s − 624·30-s − 7·31-s + 38·32-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 7.50·3-s + 5.36·5-s − 21.2·6-s + 4.91·7-s − 7.42·8-s + 91/3·9-s + 15.1·10-s + 0.904·11-s + 5.82·13-s + 13.8·14-s − 40.2·15-s − 5.75·16-s + 4.12·17-s + 85.7·18-s + 1.14·19-s − 36.8·21-s + 2.55·22-s + 0.834·23-s + 55.7·24-s + 8·25-s + 16.4·26-s − 87.5·27-s + 3.89·29-s − 113.·30-s − 1.25·31-s + 6.71·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{13} \cdot 7^{13} \cdot 127^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{13} \cdot 7^{13} \cdot 127^{13}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(26\) |
| Conductor: | \(3^{13} \cdot 7^{13} \cdot 127^{13}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.85311\times 10^{17}\) |
| Root analytic conductor: | \(4.61477\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((26,\ 3^{13} \cdot 7^{13} \cdot 127^{13} ,\ ( \ : [1/2]^{13} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(823.8414900\) |
| \(L(\frac12)\) | \(\approx\) | \(823.8414900\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T )^{13} \) |
| 7 | \( ( 1 - T )^{13} \) | |
| 127 | \( ( 1 + T )^{13} \) | |
| good | 2 | \( 1 - p^{2} T + p^{4} T^{2} - 43 T^{3} + 111 T^{4} - 119 p T^{5} + 491 T^{6} - 905 T^{7} + 1619 T^{8} - 167 p^{4} T^{9} + 539 p^{3} T^{10} - 819 p^{3} T^{11} + 153 p^{6} T^{12} - 1743 p^{3} T^{13} + 153 p^{7} T^{14} - 819 p^{5} T^{15} + 539 p^{6} T^{16} - 167 p^{8} T^{17} + 1619 p^{5} T^{18} - 905 p^{6} T^{19} + 491 p^{7} T^{20} - 119 p^{9} T^{21} + 111 p^{9} T^{22} - 43 p^{10} T^{23} + p^{15} T^{24} - p^{14} T^{25} + p^{13} T^{26} \) |
| 5 | \( 1 - 12 T + 104 T^{2} - 134 p T^{3} + 3638 T^{4} - 3392 p T^{5} + 70302 T^{6} - 262027 T^{7} + 178116 p T^{8} - 555764 p T^{9} + 8011459 T^{10} - 21421987 T^{11} + 53271576 T^{12} - 123398136 T^{13} + 53271576 p T^{14} - 21421987 p^{2} T^{15} + 8011459 p^{3} T^{16} - 555764 p^{5} T^{17} + 178116 p^{6} T^{18} - 262027 p^{6} T^{19} + 70302 p^{7} T^{20} - 3392 p^{9} T^{21} + 3638 p^{9} T^{22} - 134 p^{11} T^{23} + 104 p^{11} T^{24} - 12 p^{12} T^{25} + p^{13} T^{26} \) | |
| 11 | \( 1 - 3 T + 83 T^{2} - 21 p T^{3} + 317 p T^{4} - 8594 T^{5} + 96020 T^{6} - 206021 T^{7} + 1938774 T^{8} - 3603777 T^{9} + 30713596 T^{10} - 50136844 T^{11} + 400155761 T^{12} - 591166452 T^{13} + 400155761 p T^{14} - 50136844 p^{2} T^{15} + 30713596 p^{3} T^{16} - 3603777 p^{4} T^{17} + 1938774 p^{5} T^{18} - 206021 p^{6} T^{19} + 96020 p^{7} T^{20} - 8594 p^{8} T^{21} + 317 p^{10} T^{22} - 21 p^{11} T^{23} + 83 p^{11} T^{24} - 3 p^{12} T^{25} + p^{13} T^{26} \) | |
| 13 | \( 1 - 21 T + 303 T^{2} - 3206 T^{3} + 2173 p T^{4} - 211775 T^{5} + 1406812 T^{6} - 8366101 T^{7} + 45400042 T^{8} - 225933653 T^{9} + 1041869982 T^{10} - 4459348525 T^{11} + 17818659263 T^{12} - 66419150254 T^{13} + 17818659263 p T^{14} - 4459348525 p^{2} T^{15} + 1041869982 p^{3} T^{16} - 225933653 p^{4} T^{17} + 45400042 p^{5} T^{18} - 8366101 p^{6} T^{19} + 1406812 p^{7} T^{20} - 211775 p^{8} T^{21} + 2173 p^{10} T^{22} - 3206 p^{10} T^{23} + 303 p^{11} T^{24} - 21 p^{12} T^{25} + p^{13} T^{26} \) | |
| 17 | \( 1 - p T + 258 T^{2} - 2538 T^{3} + 1313 p T^{4} - 152819 T^{5} + 941699 T^{6} - 274768 p T^{7} + 20527283 T^{8} - 67321211 T^{9} + 167904403 T^{10} - 10092848 T^{11} - 1775763501 T^{12} + 11811409138 T^{13} - 1775763501 p T^{14} - 10092848 p^{2} T^{15} + 167904403 p^{3} T^{16} - 67321211 p^{4} T^{17} + 20527283 p^{5} T^{18} - 274768 p^{7} T^{19} + 941699 p^{7} T^{20} - 152819 p^{8} T^{21} + 1313 p^{10} T^{22} - 2538 p^{10} T^{23} + 258 p^{11} T^{24} - p^{13} T^{25} + p^{13} T^{26} \) | |
| 19 | \( 1 - 5 T + 173 T^{2} - 864 T^{3} + 14502 T^{4} - 70542 T^{5} + 782707 T^{6} - 3630922 T^{7} + 30483684 T^{8} - 132517839 T^{9} + 47863464 p T^{10} - 3646806238 T^{11} + 21484456413 T^{12} - 78131794348 T^{13} + 21484456413 p T^{14} - 3646806238 p^{2} T^{15} + 47863464 p^{4} T^{16} - 132517839 p^{4} T^{17} + 30483684 p^{5} T^{18} - 3630922 p^{6} T^{19} + 782707 p^{7} T^{20} - 70542 p^{8} T^{21} + 14502 p^{9} T^{22} - 864 p^{10} T^{23} + 173 p^{11} T^{24} - 5 p^{12} T^{25} + p^{13} T^{26} \) | |
| 23 | \( 1 - 4 T + 8 p T^{2} - 613 T^{3} + 16621 T^{4} - 45907 T^{5} + 981152 T^{6} - 2240102 T^{7} + 1851189 p T^{8} - 80788590 T^{9} + 1448692280 T^{10} - 2339155653 T^{11} + 40151933279 T^{12} - 57649846102 T^{13} + 40151933279 p T^{14} - 2339155653 p^{2} T^{15} + 1448692280 p^{3} T^{16} - 80788590 p^{4} T^{17} + 1851189 p^{6} T^{18} - 2240102 p^{6} T^{19} + 981152 p^{7} T^{20} - 45907 p^{8} T^{21} + 16621 p^{9} T^{22} - 613 p^{10} T^{23} + 8 p^{12} T^{24} - 4 p^{12} T^{25} + p^{13} T^{26} \) | |
| 29 | \( 1 - 21 T + 421 T^{2} - 5324 T^{3} + 64636 T^{4} - 613764 T^{5} + 5683567 T^{6} - 44367699 T^{7} + 342010383 T^{8} - 2302116496 T^{9} + 15447960508 T^{10} - 92186196428 T^{11} + 551990408672 T^{12} - 2963927041432 T^{13} + 551990408672 p T^{14} - 92186196428 p^{2} T^{15} + 15447960508 p^{3} T^{16} - 2302116496 p^{4} T^{17} + 342010383 p^{5} T^{18} - 44367699 p^{6} T^{19} + 5683567 p^{7} T^{20} - 613764 p^{8} T^{21} + 64636 p^{9} T^{22} - 5324 p^{10} T^{23} + 421 p^{11} T^{24} - 21 p^{12} T^{25} + p^{13} T^{26} \) | |
| 31 | \( 1 + 7 T + 7 p T^{2} + 1437 T^{3} + 23361 T^{4} + 141394 T^{5} + 1651220 T^{6} + 9026082 T^{7} + 86549696 T^{8} + 428736095 T^{9} + 3637767504 T^{10} + 16521213633 T^{11} + 129439998267 T^{12} + 546785963664 T^{13} + 129439998267 p T^{14} + 16521213633 p^{2} T^{15} + 3637767504 p^{3} T^{16} + 428736095 p^{4} T^{17} + 86549696 p^{5} T^{18} + 9026082 p^{6} T^{19} + 1651220 p^{7} T^{20} + 141394 p^{8} T^{21} + 23361 p^{9} T^{22} + 1437 p^{10} T^{23} + 7 p^{12} T^{24} + 7 p^{12} T^{25} + p^{13} T^{26} \) | |
| 37 | \( 1 - 7 T + 349 T^{2} - 2305 T^{3} + 57994 T^{4} - 369015 T^{5} + 6147489 T^{6} - 37952961 T^{7} + 468210685 T^{8} - 2784936853 T^{9} + 27265286966 T^{10} - 153098638011 T^{11} + 1255909686356 T^{12} - 6449571361272 T^{13} + 1255909686356 p T^{14} - 153098638011 p^{2} T^{15} + 27265286966 p^{3} T^{16} - 2784936853 p^{4} T^{17} + 468210685 p^{5} T^{18} - 37952961 p^{6} T^{19} + 6147489 p^{7} T^{20} - 369015 p^{8} T^{21} + 57994 p^{9} T^{22} - 2305 p^{10} T^{23} + 349 p^{11} T^{24} - 7 p^{12} T^{25} + p^{13} T^{26} \) | |
| 41 | \( 1 - 21 T + 616 T^{2} - 9402 T^{3} + 160488 T^{4} - 1938936 T^{5} + 24485336 T^{6} - 245110356 T^{7} + 2497863778 T^{8} - 21278259882 T^{9} + 182459787904 T^{10} - 1342887611296 T^{11} + 9902669942613 T^{12} - 63386892075414 T^{13} + 9902669942613 p T^{14} - 1342887611296 p^{2} T^{15} + 182459787904 p^{3} T^{16} - 21278259882 p^{4} T^{17} + 2497863778 p^{5} T^{18} - 245110356 p^{6} T^{19} + 24485336 p^{7} T^{20} - 1938936 p^{8} T^{21} + 160488 p^{9} T^{22} - 9402 p^{10} T^{23} + 616 p^{11} T^{24} - 21 p^{12} T^{25} + p^{13} T^{26} \) | |
| 43 | \( 1 + 9 T + 363 T^{2} + 2829 T^{3} + 62991 T^{4} + 434130 T^{5} + 7053472 T^{6} + 43885133 T^{7} + 580184802 T^{8} + 3304680495 T^{9} + 37503098992 T^{10} + 196157482974 T^{11} + 1966417783053 T^{12} + 9376737802572 T^{13} + 1966417783053 p T^{14} + 196157482974 p^{2} T^{15} + 37503098992 p^{3} T^{16} + 3304680495 p^{4} T^{17} + 580184802 p^{5} T^{18} + 43885133 p^{6} T^{19} + 7053472 p^{7} T^{20} + 434130 p^{8} T^{21} + 62991 p^{9} T^{22} + 2829 p^{10} T^{23} + 363 p^{11} T^{24} + 9 p^{12} T^{25} + p^{13} T^{26} \) | |
| 47 | \( 1 - 23 T + 607 T^{2} - 9855 T^{3} + 158534 T^{4} - 2026095 T^{5} + 24946377 T^{6} - 265384987 T^{7} + 2704629542 T^{8} - 24695815357 T^{9} + 215886050220 T^{10} - 1720118801318 T^{11} + 13126247577063 T^{12} - 91942347287866 T^{13} + 13126247577063 p T^{14} - 1720118801318 p^{2} T^{15} + 215886050220 p^{3} T^{16} - 24695815357 p^{4} T^{17} + 2704629542 p^{5} T^{18} - 265384987 p^{6} T^{19} + 24946377 p^{7} T^{20} - 2026095 p^{8} T^{21} + 158534 p^{9} T^{22} - 9855 p^{10} T^{23} + 607 p^{11} T^{24} - 23 p^{12} T^{25} + p^{13} T^{26} \) | |
| 53 | \( 1 - 31 T + 936 T^{2} - 18071 T^{3} + 328569 T^{4} - 4749105 T^{5} + 64916194 T^{6} - 760070261 T^{7} + 8473173808 T^{8} - 83734170303 T^{9} + 791809806079 T^{10} - 6750789985461 T^{11} + 55228222405121 T^{12} - 410273915363592 T^{13} + 55228222405121 p T^{14} - 6750789985461 p^{2} T^{15} + 791809806079 p^{3} T^{16} - 83734170303 p^{4} T^{17} + 8473173808 p^{5} T^{18} - 760070261 p^{6} T^{19} + 64916194 p^{7} T^{20} - 4749105 p^{8} T^{21} + 328569 p^{9} T^{22} - 18071 p^{10} T^{23} + 936 p^{11} T^{24} - 31 p^{12} T^{25} + p^{13} T^{26} \) | |
| 59 | \( 1 - 28 T + 748 T^{2} - 12353 T^{3} + 200007 T^{4} - 2482731 T^{5} + 31037042 T^{6} - 320236588 T^{7} + 3396643521 T^{8} - 30749512300 T^{9} + 290422351830 T^{10} - 2376472711159 T^{11} + 20434393717531 T^{12} - 152974888166578 T^{13} + 20434393717531 p T^{14} - 2376472711159 p^{2} T^{15} + 290422351830 p^{3} T^{16} - 30749512300 p^{4} T^{17} + 3396643521 p^{5} T^{18} - 320236588 p^{6} T^{19} + 31037042 p^{7} T^{20} - 2482731 p^{8} T^{21} + 200007 p^{9} T^{22} - 12353 p^{10} T^{23} + 748 p^{11} T^{24} - 28 p^{12} T^{25} + p^{13} T^{26} \) | |
| 61 | \( 1 - 29 T + 1023 T^{2} - 20439 T^{3} + 425921 T^{4} - 6588276 T^{5} + 102071192 T^{6} - 1292237308 T^{7} + 16152643052 T^{8} - 172446855471 T^{9} + 1807344944080 T^{10} - 16534828328737 T^{11} + 148161174495311 T^{12} - 1169622992942008 T^{13} + 148161174495311 p T^{14} - 16534828328737 p^{2} T^{15} + 1807344944080 p^{3} T^{16} - 172446855471 p^{4} T^{17} + 16152643052 p^{5} T^{18} - 1292237308 p^{6} T^{19} + 102071192 p^{7} T^{20} - 6588276 p^{8} T^{21} + 425921 p^{9} T^{22} - 20439 p^{10} T^{23} + 1023 p^{11} T^{24} - 29 p^{12} T^{25} + p^{13} T^{26} \) | |
| 67 | \( 1 + 18 T + 515 T^{2} + 7308 T^{3} + 124920 T^{4} + 1475761 T^{5} + 19125783 T^{6} + 196479200 T^{7} + 2123208520 T^{8} + 19682935334 T^{9} + 187612591758 T^{10} + 1616331925468 T^{11} + 14149667857895 T^{12} + 115112160627838 T^{13} + 14149667857895 p T^{14} + 1616331925468 p^{2} T^{15} + 187612591758 p^{3} T^{16} + 19682935334 p^{4} T^{17} + 2123208520 p^{5} T^{18} + 196479200 p^{6} T^{19} + 19125783 p^{7} T^{20} + 1475761 p^{8} T^{21} + 124920 p^{9} T^{22} + 7308 p^{10} T^{23} + 515 p^{11} T^{24} + 18 p^{12} T^{25} + p^{13} T^{26} \) | |
| 71 | \( 1 - 10 T + 556 T^{2} - 4573 T^{3} + 149491 T^{4} - 1052465 T^{5} + 26282634 T^{6} - 161622081 T^{7} + 3414944265 T^{8} - 18615640006 T^{9} + 350734762938 T^{10} - 1720351297082 T^{11} + 29653858976227 T^{12} - 132764914433646 T^{13} + 29653858976227 p T^{14} - 1720351297082 p^{2} T^{15} + 350734762938 p^{3} T^{16} - 18615640006 p^{4} T^{17} + 3414944265 p^{5} T^{18} - 161622081 p^{6} T^{19} + 26282634 p^{7} T^{20} - 1052465 p^{8} T^{21} + 149491 p^{9} T^{22} - 4573 p^{10} T^{23} + 556 p^{11} T^{24} - 10 p^{12} T^{25} + p^{13} T^{26} \) | |
| 73 | \( 1 - 24 T + 870 T^{2} - 15387 T^{3} + 325681 T^{4} - 4669441 T^{5} + 74019844 T^{6} - 905055656 T^{7} + 11773114747 T^{8} - 126039249920 T^{9} + 1404528656478 T^{10} - 13325817517321 T^{11} + 130094309435051 T^{12} - 1097731775448310 T^{13} + 130094309435051 p T^{14} - 13325817517321 p^{2} T^{15} + 1404528656478 p^{3} T^{16} - 126039249920 p^{4} T^{17} + 11773114747 p^{5} T^{18} - 905055656 p^{6} T^{19} + 74019844 p^{7} T^{20} - 4669441 p^{8} T^{21} + 325681 p^{9} T^{22} - 15387 p^{10} T^{23} + 870 p^{11} T^{24} - 24 p^{12} T^{25} + p^{13} T^{26} \) | |
| 79 | \( 1 + 28 T + 1025 T^{2} + 19014 T^{3} + 398721 T^{4} + 5420565 T^{5} + 81531672 T^{6} + 832892678 T^{7} + 9725170712 T^{8} + 72303138705 T^{9} + 699331712541 T^{10} + 3412737965740 T^{11} + 35417479052632 T^{12} + 139160460017484 T^{13} + 35417479052632 p T^{14} + 3412737965740 p^{2} T^{15} + 699331712541 p^{3} T^{16} + 72303138705 p^{4} T^{17} + 9725170712 p^{5} T^{18} + 832892678 p^{6} T^{19} + 81531672 p^{7} T^{20} + 5420565 p^{8} T^{21} + 398721 p^{9} T^{22} + 19014 p^{10} T^{23} + 1025 p^{11} T^{24} + 28 p^{12} T^{25} + p^{13} T^{26} \) | |
| 83 | \( 1 - 26 T + 912 T^{2} - 16870 T^{3} + 351403 T^{4} - 5167744 T^{5} + 81355420 T^{6} - 1007182561 T^{7} + 13151871421 T^{8} - 142231819988 T^{9} + 1621508875722 T^{10} - 15730315041009 T^{11} + 161523128267333 T^{12} - 1428718611245444 T^{13} + 161523128267333 p T^{14} - 15730315041009 p^{2} T^{15} + 1621508875722 p^{3} T^{16} - 142231819988 p^{4} T^{17} + 13151871421 p^{5} T^{18} - 1007182561 p^{6} T^{19} + 81355420 p^{7} T^{20} - 5167744 p^{8} T^{21} + 351403 p^{9} T^{22} - 16870 p^{10} T^{23} + 912 p^{11} T^{24} - 26 p^{12} T^{25} + p^{13} T^{26} \) | |
| 89 | \( 1 - 44 T + 1789 T^{2} - 48025 T^{3} + 1179457 T^{4} - 23460109 T^{5} + 431592848 T^{6} - 6880773166 T^{7} + 102377806578 T^{8} - 1360607322250 T^{9} + 16975488400752 T^{10} - 191986074688377 T^{11} + 2045012097441543 T^{12} - 19868978401857274 T^{13} + 2045012097441543 p T^{14} - 191986074688377 p^{2} T^{15} + 16975488400752 p^{3} T^{16} - 1360607322250 p^{4} T^{17} + 102377806578 p^{5} T^{18} - 6880773166 p^{6} T^{19} + 431592848 p^{7} T^{20} - 23460109 p^{8} T^{21} + 1179457 p^{9} T^{22} - 48025 p^{10} T^{23} + 1789 p^{11} T^{24} - 44 p^{12} T^{25} + p^{13} T^{26} \) | |
| 97 | \( 1 - 17 T + 998 T^{2} - 14430 T^{3} + 471225 T^{4} - 5972993 T^{5} + 141228807 T^{6} - 1591368836 T^{7} + 30095354665 T^{8} - 303043084483 T^{9} + 4820083155707 T^{10} - 43368648096954 T^{11} + 597070384172585 T^{12} - 4777592162764534 T^{13} + 597070384172585 p T^{14} - 43368648096954 p^{2} T^{15} + 4820083155707 p^{3} T^{16} - 303043084483 p^{4} T^{17} + 30095354665 p^{5} T^{18} - 1591368836 p^{6} T^{19} + 141228807 p^{7} T^{20} - 5972993 p^{8} T^{21} + 471225 p^{9} T^{22} - 14430 p^{10} T^{23} + 998 p^{11} T^{24} - 17 p^{12} T^{25} + p^{13} T^{26} \) | |
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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
−2.38370692901848216671501614382, −2.28603323537416401569130056584, −2.22880751558630031144177133022, −2.22618674916304198277685535928, −2.11161221417565413023267787836, −1.97193061025269719249023388994, −1.87573168663846264414950532925, −1.84369523070955088628243953521, −1.77323115692074928506214371515, −1.66225603332126532562437101329, −1.56209068411245040481810359703, −1.55341916522847542101597656140, −1.53863349465926401733710499210, −1.33914781640161568491874195668, −1.11484293217935441864507168165, −1.07914017326277464235941852287, −1.04300782121475259797288442401, −0.881922603253423503105043709001, −0.878663701037864219906806491298, −0.797195190014218018566404141771, −0.72497196903688540661232156372, −0.62642269402887886976725523056, −0.60599546733979404406229075577, −0.59643922799407332513135805917, −0.59074122922258193058379353128, 0.59074122922258193058379353128, 0.59643922799407332513135805917, 0.60599546733979404406229075577, 0.62642269402887886976725523056, 0.72497196903688540661232156372, 0.797195190014218018566404141771, 0.878663701037864219906806491298, 0.881922603253423503105043709001, 1.04300782121475259797288442401, 1.07914017326277464235941852287, 1.11484293217935441864507168165, 1.33914781640161568491874195668, 1.53863349465926401733710499210, 1.55341916522847542101597656140, 1.56209068411245040481810359703, 1.66225603332126532562437101329, 1.77323115692074928506214371515, 1.84369523070955088628243953521, 1.87573168663846264414950532925, 1.97193061025269719249023388994, 2.11161221417565413023267787836, 2.22618674916304198277685535928, 2.22880751558630031144177133022, 2.28603323537416401569130056584, 2.38370692901848216671501614382
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.