Dirichlet series
| L(s) = 1 | + 2·2-s + 5·3-s − 4·4-s + 11·5-s + 10·6-s + 7·7-s − 14·8-s − 2·9-s + 22·10-s + 4·11-s − 20·12-s + 14·13-s + 14·14-s + 55·15-s − 5·16-s + 19·17-s − 4·18-s + 12·19-s − 44·20-s + 35·21-s + 8·22-s − 23-s − 70·24-s + 28·25-s + 28·26-s − 54·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.88·3-s − 2·4-s + 4.91·5-s + 4.08·6-s + 2.64·7-s − 4.94·8-s − 2/3·9-s + 6.95·10-s + 1.20·11-s − 5.77·12-s + 3.88·13-s + 3.74·14-s + 14.2·15-s − 5/4·16-s + 4.60·17-s − 0.942·18-s + 2.75·19-s − 9.83·20-s + 7.63·21-s + 1.70·22-s − 0.208·23-s − 14.2·24-s + 28/5·25-s + 5.49·26-s − 10.3·27-s − 5.29·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{19} \cdot 43^{19}\right)^{s/2} \, \Gamma_{\C}(s)^{19} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{19} \cdot 43^{19}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{19} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(38\) |
| Conductor: | \(17^{19} \cdot 43^{19}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.61156\times 10^{14}\) |
| Root analytic conductor: | \(2.41600\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((38,\ 17^{19} \cdot 43^{19} ,\ ( \ : [1/2]^{19} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(994.2596726\) |
| \(L(\frac12)\) | \(\approx\) | \(994.2596726\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 17 | \( ( 1 - T )^{19} \) |
| 43 | \( ( 1 + T )^{19} \) | |
| good | 2 | \( 1 - p T + p^{3} T^{2} - 5 p T^{3} + 29 T^{4} - 13 p T^{5} + 87 T^{6} - 71 T^{7} + 131 p T^{8} - 189 T^{9} + 681 T^{10} - 399 T^{11} + 99 p^{4} T^{12} - 779 T^{13} + 443 p^{3} T^{14} - 375 p^{2} T^{15} + 471 p^{4} T^{16} - 693 p^{2} T^{17} + 959 p^{4} T^{18} - 83 p^{6} T^{19} + 959 p^{5} T^{20} - 693 p^{4} T^{21} + 471 p^{7} T^{22} - 375 p^{6} T^{23} + 443 p^{8} T^{24} - 779 p^{6} T^{25} + 99 p^{11} T^{26} - 399 p^{8} T^{27} + 681 p^{9} T^{28} - 189 p^{10} T^{29} + 131 p^{12} T^{30} - 71 p^{12} T^{31} + 87 p^{13} T^{32} - 13 p^{15} T^{33} + 29 p^{15} T^{34} - 5 p^{17} T^{35} + p^{20} T^{36} - p^{19} T^{37} + p^{19} T^{38} \) |
| 3 | \( 1 - 5 T + p^{3} T^{2} - 91 T^{3} + 109 p T^{4} - 917 T^{5} + 2692 T^{6} - 6737 T^{7} + 5825 p T^{8} - 40382 T^{9} + 95497 T^{10} - 206258 T^{11} + 150443 p T^{12} - 917101 T^{13} + 625604 p T^{14} - 3608699 T^{15} + 6962041 T^{16} - 12706865 T^{17} + 7735880 p T^{18} - 4467610 p^{2} T^{19} + 7735880 p^{2} T^{20} - 12706865 p^{2} T^{21} + 6962041 p^{3} T^{22} - 3608699 p^{4} T^{23} + 625604 p^{6} T^{24} - 917101 p^{6} T^{25} + 150443 p^{8} T^{26} - 206258 p^{8} T^{27} + 95497 p^{9} T^{28} - 40382 p^{10} T^{29} + 5825 p^{12} T^{30} - 6737 p^{12} T^{31} + 2692 p^{13} T^{32} - 917 p^{14} T^{33} + 109 p^{16} T^{34} - 91 p^{16} T^{35} + p^{20} T^{36} - 5 p^{18} T^{37} + p^{19} T^{38} \) | |
| 5 | \( 1 - 11 T + 93 T^{2} - 577 T^{3} + 3112 T^{4} - 578 p^{2} T^{5} + 12218 p T^{6} - 234848 T^{7} + 843343 T^{8} - 2830713 T^{9} + 9018157 T^{10} - 27287597 T^{11} + 79155767 T^{12} - 220152492 T^{13} + 118096348 p T^{14} - 1526714046 T^{15} + 763917176 p T^{16} - 1847898627 p T^{17} + 21665733632 T^{18} - 49190638264 T^{19} + 21665733632 p T^{20} - 1847898627 p^{3} T^{21} + 763917176 p^{4} T^{22} - 1526714046 p^{4} T^{23} + 118096348 p^{6} T^{24} - 220152492 p^{6} T^{25} + 79155767 p^{7} T^{26} - 27287597 p^{8} T^{27} + 9018157 p^{9} T^{28} - 2830713 p^{10} T^{29} + 843343 p^{11} T^{30} - 234848 p^{12} T^{31} + 12218 p^{14} T^{32} - 578 p^{16} T^{33} + 3112 p^{15} T^{34} - 577 p^{16} T^{35} + 93 p^{17} T^{36} - 11 p^{18} T^{37} + p^{19} T^{38} \) | |
| 7 | \( 1 - p T + 68 T^{2} - 358 T^{3} + 2152 T^{4} - 1375 p T^{5} + 6428 p T^{6} - 178732 T^{7} + 707997 T^{8} - 2559961 T^{9} + 1286540 p T^{10} - 30060138 T^{11} + 13830447 p T^{12} - 301227093 T^{13} + 908088082 T^{14} - 2656056828 T^{15} + 7611743644 T^{16} - 21110762354 T^{17} + 58026267877 T^{18} - 21978666480 p T^{19} + 58026267877 p T^{20} - 21110762354 p^{2} T^{21} + 7611743644 p^{3} T^{22} - 2656056828 p^{4} T^{23} + 908088082 p^{5} T^{24} - 301227093 p^{6} T^{25} + 13830447 p^{8} T^{26} - 30060138 p^{8} T^{27} + 1286540 p^{10} T^{28} - 2559961 p^{10} T^{29} + 707997 p^{11} T^{30} - 178732 p^{12} T^{31} + 6428 p^{14} T^{32} - 1375 p^{15} T^{33} + 2152 p^{15} T^{34} - 358 p^{16} T^{35} + 68 p^{17} T^{36} - p^{19} T^{37} + p^{19} T^{38} \) | |
| 11 | \( 1 - 4 T + 78 T^{2} - 240 T^{3} + 3138 T^{4} - 826 p T^{5} + 8376 p T^{6} - 267556 T^{7} + 2135091 T^{8} - 582442 p T^{9} + 41313938 T^{10} - 130622120 T^{11} + 691389286 T^{12} - 2289844938 T^{13} + 10204054431 T^{14} - 34927323484 T^{15} + 135456031791 T^{16} - 466649085542 T^{17} + 1631244109576 T^{18} - 5470524915536 T^{19} + 1631244109576 p T^{20} - 466649085542 p^{2} T^{21} + 135456031791 p^{3} T^{22} - 34927323484 p^{4} T^{23} + 10204054431 p^{5} T^{24} - 2289844938 p^{6} T^{25} + 691389286 p^{7} T^{26} - 130622120 p^{8} T^{27} + 41313938 p^{9} T^{28} - 582442 p^{11} T^{29} + 2135091 p^{11} T^{30} - 267556 p^{12} T^{31} + 8376 p^{14} T^{32} - 826 p^{15} T^{33} + 3138 p^{15} T^{34} - 240 p^{16} T^{35} + 78 p^{17} T^{36} - 4 p^{18} T^{37} + p^{19} T^{38} \) | |
| 13 | \( 1 - 14 T + 214 T^{2} - 152 p T^{3} + 18098 T^{4} - 9795 p T^{5} + 874435 T^{6} - 4998199 T^{7} + 27866719 T^{8} - 133523489 T^{9} + 626684285 T^{10} - 2545522055 T^{11} + 10205711199 T^{12} - 2678789103 p T^{13} + 119106101269 T^{14} - 327318106617 T^{15} + 947708897568 T^{16} - 1896517306490 T^{17} + 5393620989941 T^{18} - 10243698755962 T^{19} + 5393620989941 p T^{20} - 1896517306490 p^{2} T^{21} + 947708897568 p^{3} T^{22} - 327318106617 p^{4} T^{23} + 119106101269 p^{5} T^{24} - 2678789103 p^{7} T^{25} + 10205711199 p^{7} T^{26} - 2545522055 p^{8} T^{27} + 626684285 p^{9} T^{28} - 133523489 p^{10} T^{29} + 27866719 p^{11} T^{30} - 4998199 p^{12} T^{31} + 874435 p^{13} T^{32} - 9795 p^{15} T^{33} + 18098 p^{15} T^{34} - 152 p^{17} T^{35} + 214 p^{17} T^{36} - 14 p^{18} T^{37} + p^{19} T^{38} \) | |
| 19 | \( 1 - 12 T + 231 T^{2} - 2046 T^{3} + 23162 T^{4} - 166538 T^{5} + 1418269 T^{6} - 8744092 T^{7} + 61566152 T^{8} - 339004010 T^{9} + 2080721747 T^{10} - 10620093376 T^{11} + 58951261327 T^{12} - 288096395550 T^{13} + 1482678698100 T^{14} - 7059263953764 T^{15} + 34080082608516 T^{16} - 157553616884706 T^{17} + 715767251171667 T^{18} - 3166355459806596 T^{19} + 715767251171667 p T^{20} - 157553616884706 p^{2} T^{21} + 34080082608516 p^{3} T^{22} - 7059263953764 p^{4} T^{23} + 1482678698100 p^{5} T^{24} - 288096395550 p^{6} T^{25} + 58951261327 p^{7} T^{26} - 10620093376 p^{8} T^{27} + 2080721747 p^{9} T^{28} - 339004010 p^{10} T^{29} + 61566152 p^{11} T^{30} - 8744092 p^{12} T^{31} + 1418269 p^{13} T^{32} - 166538 p^{14} T^{33} + 23162 p^{15} T^{34} - 2046 p^{16} T^{35} + 231 p^{17} T^{36} - 12 p^{18} T^{37} + p^{19} T^{38} \) | |
| 23 | \( 1 + T + 169 T^{2} - 12 T^{3} + 13946 T^{4} - 16196 T^{5} + 765929 T^{6} - 1729078 T^{7} + 32357388 T^{8} - 106127027 T^{9} + 1157314113 T^{10} - 4683814572 T^{11} + 37590860305 T^{12} - 163415289321 T^{13} + 1148017887760 T^{14} - 4790799294058 T^{15} + 32712484129236 T^{16} - 124149907712945 T^{17} + 845806190189449 T^{18} - 2956351382217424 T^{19} + 845806190189449 p T^{20} - 124149907712945 p^{2} T^{21} + 32712484129236 p^{3} T^{22} - 4790799294058 p^{4} T^{23} + 1148017887760 p^{5} T^{24} - 163415289321 p^{6} T^{25} + 37590860305 p^{7} T^{26} - 4683814572 p^{8} T^{27} + 1157314113 p^{9} T^{28} - 106127027 p^{10} T^{29} + 32357388 p^{11} T^{30} - 1729078 p^{12} T^{31} + 765929 p^{13} T^{32} - 16196 p^{14} T^{33} + 13946 p^{15} T^{34} - 12 p^{16} T^{35} + 169 p^{17} T^{36} + p^{18} T^{37} + p^{19} T^{38} \) | |
| 29 | \( 1 - 41 T + 1089 T^{2} - 21265 T^{3} + 341538 T^{4} - 4672957 T^{5} + 56430977 T^{6} - 611476694 T^{7} + 6047148638 T^{8} - 1900078336 p T^{9} + 467277963785 T^{10} - 3710864309128 T^{11} + 958028018087 p T^{12} - 196970363235912 T^{13} + 1328790720935966 T^{14} - 8557325803031210 T^{15} + 52792953673230190 T^{16} - 312660111943143098 T^{17} + 1781512170633487173 T^{18} - 9773547813493035982 T^{19} + 1781512170633487173 p T^{20} - 312660111943143098 p^{2} T^{21} + 52792953673230190 p^{3} T^{22} - 8557325803031210 p^{4} T^{23} + 1328790720935966 p^{5} T^{24} - 196970363235912 p^{6} T^{25} + 958028018087 p^{8} T^{26} - 3710864309128 p^{8} T^{27} + 467277963785 p^{9} T^{28} - 1900078336 p^{11} T^{29} + 6047148638 p^{11} T^{30} - 611476694 p^{12} T^{31} + 56430977 p^{13} T^{32} - 4672957 p^{14} T^{33} + 341538 p^{15} T^{34} - 21265 p^{16} T^{35} + 1089 p^{17} T^{36} - 41 p^{18} T^{37} + p^{19} T^{38} \) | |
| 31 | \( 1 + 8 T + 278 T^{2} + 2240 T^{3} + 42049 T^{4} + 327960 T^{5} + 4430105 T^{6} + 32923176 T^{7} + 359367120 T^{8} + 2525567424 T^{9} + 23638859341 T^{10} + 156677968944 T^{11} + 1301358545845 T^{12} + 8125591802592 T^{13} + 61175290385916 T^{14} + 359506047750616 T^{15} + 2487501062988676 T^{16} + 13736067246740544 T^{17} + 88159141553938611 T^{18} + 456234699212299808 T^{19} + 88159141553938611 p T^{20} + 13736067246740544 p^{2} T^{21} + 2487501062988676 p^{3} T^{22} + 359506047750616 p^{4} T^{23} + 61175290385916 p^{5} T^{24} + 8125591802592 p^{6} T^{25} + 1301358545845 p^{7} T^{26} + 156677968944 p^{8} T^{27} + 23638859341 p^{9} T^{28} + 2525567424 p^{10} T^{29} + 359367120 p^{11} T^{30} + 32923176 p^{12} T^{31} + 4430105 p^{13} T^{32} + 327960 p^{14} T^{33} + 42049 p^{15} T^{34} + 2240 p^{16} T^{35} + 278 p^{17} T^{36} + 8 p^{18} T^{37} + p^{19} T^{38} \) | |
| 37 | \( 1 - 50 T + 1425 T^{2} - 29461 T^{3} + 490242 T^{4} - 6918426 T^{5} + 85558550 T^{6} - 947084225 T^{7} + 9530614305 T^{8} - 88198177881 T^{9} + 757668183283 T^{10} - 6089525403205 T^{11} + 46121901280583 T^{12} - 331463010330233 T^{13} + 2276400123202178 T^{14} - 15050682901348850 T^{15} + 96551260049863866 T^{16} - 16368336709697023 p T^{17} + 3740293211863341492 T^{18} - 22853121352986233194 T^{19} + 3740293211863341492 p T^{20} - 16368336709697023 p^{3} T^{21} + 96551260049863866 p^{3} T^{22} - 15050682901348850 p^{4} T^{23} + 2276400123202178 p^{5} T^{24} - 331463010330233 p^{6} T^{25} + 46121901280583 p^{7} T^{26} - 6089525403205 p^{8} T^{27} + 757668183283 p^{9} T^{28} - 88198177881 p^{10} T^{29} + 9530614305 p^{11} T^{30} - 947084225 p^{12} T^{31} + 85558550 p^{13} T^{32} - 6918426 p^{14} T^{33} + 490242 p^{15} T^{34} - 29461 p^{16} T^{35} + 1425 p^{17} T^{36} - 50 p^{18} T^{37} + p^{19} T^{38} \) | |
| 41 | \( 1 - 6 T + 329 T^{2} - 2597 T^{3} + 57459 T^{4} - 524828 T^{5} + 7252044 T^{6} - 68508618 T^{7} + 735498471 T^{8} - 6654299634 T^{9} + 62133310512 T^{10} - 520364734448 T^{11} + 4436651692828 T^{12} - 34312870956454 T^{13} + 270999727686889 T^{14} - 1956594830205334 T^{15} + 14343311242892507 T^{16} - 97569318390178278 T^{17} + 665685697462340656 T^{18} - 4269431477509522454 T^{19} + 665685697462340656 p T^{20} - 97569318390178278 p^{2} T^{21} + 14343311242892507 p^{3} T^{22} - 1956594830205334 p^{4} T^{23} + 270999727686889 p^{5} T^{24} - 34312870956454 p^{6} T^{25} + 4436651692828 p^{7} T^{26} - 520364734448 p^{8} T^{27} + 62133310512 p^{9} T^{28} - 6654299634 p^{10} T^{29} + 735498471 p^{11} T^{30} - 68508618 p^{12} T^{31} + 7252044 p^{13} T^{32} - 524828 p^{14} T^{33} + 57459 p^{15} T^{34} - 2597 p^{16} T^{35} + 329 p^{17} T^{36} - 6 p^{18} T^{37} + p^{19} T^{38} \) | |
| 47 | \( 1 + 21 T + 677 T^{2} + 10969 T^{3} + 211054 T^{4} + 2844100 T^{5} + 41548248 T^{6} + 484263554 T^{7} + 5869739497 T^{8} + 60648007833 T^{9} + 638194889529 T^{10} + 5947583495067 T^{11} + 55859565574255 T^{12} + 475815613636288 T^{13} + 4065146628527674 T^{14} + 31985422825539390 T^{15} + 252021018148047428 T^{16} + 1846196541972732625 T^{17} + 13541484727674359744 T^{18} + 92772516716756121618 T^{19} + 13541484727674359744 p T^{20} + 1846196541972732625 p^{2} T^{21} + 252021018148047428 p^{3} T^{22} + 31985422825539390 p^{4} T^{23} + 4065146628527674 p^{5} T^{24} + 475815613636288 p^{6} T^{25} + 55859565574255 p^{7} T^{26} + 5947583495067 p^{8} T^{27} + 638194889529 p^{9} T^{28} + 60648007833 p^{10} T^{29} + 5869739497 p^{11} T^{30} + 484263554 p^{12} T^{31} + 41548248 p^{13} T^{32} + 2844100 p^{14} T^{33} + 211054 p^{15} T^{34} + 10969 p^{16} T^{35} + 677 p^{17} T^{36} + 21 p^{18} T^{37} + p^{19} T^{38} \) | |
| 53 | \( 1 + 9 T + 780 T^{2} + 7191 T^{3} + 298306 T^{4} + 2747179 T^{5} + 74271353 T^{6} + 669370235 T^{7} + 13470997987 T^{8} + 116907469488 T^{9} + 1887197472827 T^{10} + 15574704721246 T^{11} + 211369128178713 T^{12} + 1642619886604935 T^{13} + 19338636960384239 T^{14} + 140366531844670077 T^{15} + 1465133811805399956 T^{16} + 9857725468710025997 T^{17} + 92664867271402773573 T^{18} + \)\(57\!\cdots\!52\)\( T^{19} + 92664867271402773573 p T^{20} + 9857725468710025997 p^{2} T^{21} + 1465133811805399956 p^{3} T^{22} + 140366531844670077 p^{4} T^{23} + 19338636960384239 p^{5} T^{24} + 1642619886604935 p^{6} T^{25} + 211369128178713 p^{7} T^{26} + 15574704721246 p^{8} T^{27} + 1887197472827 p^{9} T^{28} + 116907469488 p^{10} T^{29} + 13470997987 p^{11} T^{30} + 669370235 p^{12} T^{31} + 74271353 p^{13} T^{32} + 2747179 p^{14} T^{33} + 298306 p^{15} T^{34} + 7191 p^{16} T^{35} + 780 p^{17} T^{36} + 9 p^{18} T^{37} + p^{19} T^{38} \) | |
| 59 | \( 1 + 4 T + 663 T^{2} + 3080 T^{3} + 223947 T^{4} + 1136719 T^{5} + 50942212 T^{6} + 271555351 T^{7} + 8709381803 T^{8} + 47376294771 T^{9} + 1184651680507 T^{10} + 6430775254801 T^{11} + 132552253794403 T^{12} + 704974438872715 T^{13} + 12456364724326576 T^{14} + 63869633975156229 T^{15} + 16881347987594659 p T^{16} + 4849718465975753852 T^{17} + 68286925881903116430 T^{18} + \)\(31\!\cdots\!80\)\( T^{19} + 68286925881903116430 p T^{20} + 4849718465975753852 p^{2} T^{21} + 16881347987594659 p^{4} T^{22} + 63869633975156229 p^{4} T^{23} + 12456364724326576 p^{5} T^{24} + 704974438872715 p^{6} T^{25} + 132552253794403 p^{7} T^{26} + 6430775254801 p^{8} T^{27} + 1184651680507 p^{9} T^{28} + 47376294771 p^{10} T^{29} + 8709381803 p^{11} T^{30} + 271555351 p^{12} T^{31} + 50942212 p^{13} T^{32} + 1136719 p^{14} T^{33} + 223947 p^{15} T^{34} + 3080 p^{16} T^{35} + 663 p^{17} T^{36} + 4 p^{18} T^{37} + p^{19} T^{38} \) | |
| 61 | \( 1 - 68 T + 2772 T^{2} - 83142 T^{3} + 2030228 T^{4} - 42331623 T^{5} + 778239125 T^{6} - 12870365339 T^{7} + 194349423761 T^{8} - 2708043949855 T^{9} + 35106902577791 T^{10} - 426061706903635 T^{11} + 4864687884328335 T^{12} - 52455906350851769 T^{13} + 535835788019690917 T^{14} - 5197409100103617711 T^{15} + 47958733074763543974 T^{16} - \)\(42\!\cdots\!88\)\( T^{17} + 57919315019313734347 p T^{18} - \)\(28\!\cdots\!46\)\( T^{19} + 57919315019313734347 p^{2} T^{20} - \)\(42\!\cdots\!88\)\( p^{2} T^{21} + 47958733074763543974 p^{3} T^{22} - 5197409100103617711 p^{4} T^{23} + 535835788019690917 p^{5} T^{24} - 52455906350851769 p^{6} T^{25} + 4864687884328335 p^{7} T^{26} - 426061706903635 p^{8} T^{27} + 35106902577791 p^{9} T^{28} - 2708043949855 p^{10} T^{29} + 194349423761 p^{11} T^{30} - 12870365339 p^{12} T^{31} + 778239125 p^{13} T^{32} - 42331623 p^{14} T^{33} + 2030228 p^{15} T^{34} - 83142 p^{16} T^{35} + 2772 p^{17} T^{36} - 68 p^{18} T^{37} + p^{19} T^{38} \) | |
| 67 | \( 1 + 714 T^{2} + 1204 T^{3} + 255028 T^{4} + 804905 T^{5} + 61253999 T^{6} + 265709657 T^{7} + 11164484743 T^{8} + 57803704861 T^{9} + 1642269774787 T^{10} + 9320581662205 T^{11} + 201525232481375 T^{12} + 1185203274545387 T^{13} + 21009811816113565 T^{14} + 123133838415160821 T^{15} + 1880602591764184612 T^{16} + 10662718143077767072 T^{17} + \)\(14\!\cdots\!91\)\( T^{18} + \)\(77\!\cdots\!28\)\( T^{19} + \)\(14\!\cdots\!91\)\( p T^{20} + 10662718143077767072 p^{2} T^{21} + 1880602591764184612 p^{3} T^{22} + 123133838415160821 p^{4} T^{23} + 21009811816113565 p^{5} T^{24} + 1185203274545387 p^{6} T^{25} + 201525232481375 p^{7} T^{26} + 9320581662205 p^{8} T^{27} + 1642269774787 p^{9} T^{28} + 57803704861 p^{10} T^{29} + 11164484743 p^{11} T^{30} + 265709657 p^{12} T^{31} + 61253999 p^{13} T^{32} + 804905 p^{14} T^{33} + 255028 p^{15} T^{34} + 1204 p^{16} T^{35} + 714 p^{17} T^{36} + p^{19} T^{38} \) | |
| 71 | \( 1 - 23 T + 959 T^{2} - 16971 T^{3} + 413122 T^{4} - 6037064 T^{5} + 110513458 T^{6} - 1388497956 T^{7} + 21048938021 T^{8} - 233341820817 T^{9} + 3083209420571 T^{10} - 30731327371139 T^{11} + 365846041395945 T^{12} - 3330750195343332 T^{13} + 36606915100315392 T^{14} - 308966391422238732 T^{15} + 3193863140318323868 T^{16} - 25346303937904705273 T^{17} + \)\(24\!\cdots\!96\)\( T^{18} - 26518419160040136906 p T^{19} + \)\(24\!\cdots\!96\)\( p T^{20} - 25346303937904705273 p^{2} T^{21} + 3193863140318323868 p^{3} T^{22} - 308966391422238732 p^{4} T^{23} + 36606915100315392 p^{5} T^{24} - 3330750195343332 p^{6} T^{25} + 365846041395945 p^{7} T^{26} - 30731327371139 p^{8} T^{27} + 3083209420571 p^{9} T^{28} - 233341820817 p^{10} T^{29} + 21048938021 p^{11} T^{30} - 1388497956 p^{12} T^{31} + 110513458 p^{13} T^{32} - 6037064 p^{14} T^{33} + 413122 p^{15} T^{34} - 16971 p^{16} T^{35} + 959 p^{17} T^{36} - 23 p^{18} T^{37} + p^{19} T^{38} \) | |
| 73 | \( 1 + T + 155 T^{2} - 525 T^{3} + 15396 T^{4} - 76526 T^{5} + 1536042 T^{6} - 5489394 T^{7} + 163341765 T^{8} - 332345851 T^{9} + 14705850595 T^{10} - 48179872195 T^{11} + 1287269814421 T^{12} - 4674939681756 T^{13} + 114439288379406 T^{14} - 354784885197044 T^{15} + 9318036949431740 T^{16} - 25780620818805041 T^{17} + 650676747902826484 T^{18} - 2222829081672155084 T^{19} + 650676747902826484 p T^{20} - 25780620818805041 p^{2} T^{21} + 9318036949431740 p^{3} T^{22} - 354784885197044 p^{4} T^{23} + 114439288379406 p^{5} T^{24} - 4674939681756 p^{6} T^{25} + 1287269814421 p^{7} T^{26} - 48179872195 p^{8} T^{27} + 14705850595 p^{9} T^{28} - 332345851 p^{10} T^{29} + 163341765 p^{11} T^{30} - 5489394 p^{12} T^{31} + 1536042 p^{13} T^{32} - 76526 p^{14} T^{33} + 15396 p^{15} T^{34} - 525 p^{16} T^{35} + 155 p^{17} T^{36} + p^{18} T^{37} + p^{19} T^{38} \) | |
| 79 | \( 1 - 16 T + 1088 T^{2} - 16598 T^{3} + 587745 T^{4} - 8421099 T^{5} + 208411191 T^{6} - 2779796750 T^{7} + 54166537426 T^{8} - 669532456249 T^{9} + 10933776008115 T^{10} - 1582336687804 p T^{11} + 1774608475298123 T^{12} - 18752935443267731 T^{13} + 236771609202101054 T^{14} - 2311432296886472594 T^{15} + 26340351491274731314 T^{16} - \)\(23\!\cdots\!85\)\( T^{17} + \)\(24\!\cdots\!29\)\( T^{18} - \)\(20\!\cdots\!60\)\( T^{19} + \)\(24\!\cdots\!29\)\( p T^{20} - \)\(23\!\cdots\!85\)\( p^{2} T^{21} + 26340351491274731314 p^{3} T^{22} - 2311432296886472594 p^{4} T^{23} + 236771609202101054 p^{5} T^{24} - 18752935443267731 p^{6} T^{25} + 1774608475298123 p^{7} T^{26} - 1582336687804 p^{9} T^{27} + 10933776008115 p^{9} T^{28} - 669532456249 p^{10} T^{29} + 54166537426 p^{11} T^{30} - 2779796750 p^{12} T^{31} + 208411191 p^{13} T^{32} - 8421099 p^{14} T^{33} + 587745 p^{15} T^{34} - 16598 p^{16} T^{35} + 1088 p^{17} T^{36} - 16 p^{18} T^{37} + p^{19} T^{38} \) | |
| 83 | \( 1 + 32 T + 1294 T^{2} + 27440 T^{3} + 652176 T^{4} + 10412368 T^{5} + 183774856 T^{6} + 2329424740 T^{7} + 33515504631 T^{8} + 345717824698 T^{9} + 4346790251284 T^{10} + 37329806752684 T^{11} + 449919524882637 T^{12} + 3418622112545116 T^{13} + 44104241519868560 T^{14} + 328985180736003984 T^{15} + 4537091695313368740 T^{16} + 33937149276195803290 T^{17} + \)\(44\!\cdots\!87\)\( T^{18} + \)\(31\!\cdots\!28\)\( T^{19} + \)\(44\!\cdots\!87\)\( p T^{20} + 33937149276195803290 p^{2} T^{21} + 4537091695313368740 p^{3} T^{22} + 328985180736003984 p^{4} T^{23} + 44104241519868560 p^{5} T^{24} + 3418622112545116 p^{6} T^{25} + 449919524882637 p^{7} T^{26} + 37329806752684 p^{8} T^{27} + 4346790251284 p^{9} T^{28} + 345717824698 p^{10} T^{29} + 33515504631 p^{11} T^{30} + 2329424740 p^{12} T^{31} + 183774856 p^{13} T^{32} + 10412368 p^{14} T^{33} + 652176 p^{15} T^{34} + 27440 p^{16} T^{35} + 1294 p^{17} T^{36} + 32 p^{18} T^{37} + p^{19} T^{38} \) | |
| 89 | \( 1 - 11 T + 1029 T^{2} - 10419 T^{3} + 514819 T^{4} - 4787456 T^{5} + 167708504 T^{6} - 1431683200 T^{7} + 40205840919 T^{8} - 315510651859 T^{9} + 7598628297088 T^{10} - 54970558604900 T^{11} + 1183113403731972 T^{12} - 7921297255788585 T^{13} + 156379644648551281 T^{14} - 973614617753864192 T^{15} + 17913594846055336327 T^{16} - \)\(10\!\cdots\!69\)\( T^{17} + \)\(18\!\cdots\!24\)\( T^{18} - \)\(98\!\cdots\!74\)\( T^{19} + \)\(18\!\cdots\!24\)\( p T^{20} - \)\(10\!\cdots\!69\)\( p^{2} T^{21} + 17913594846055336327 p^{3} T^{22} - 973614617753864192 p^{4} T^{23} + 156379644648551281 p^{5} T^{24} - 7921297255788585 p^{6} T^{25} + 1183113403731972 p^{7} T^{26} - 54970558604900 p^{8} T^{27} + 7598628297088 p^{9} T^{28} - 315510651859 p^{10} T^{29} + 40205840919 p^{11} T^{30} - 1431683200 p^{12} T^{31} + 167708504 p^{13} T^{32} - 4787456 p^{14} T^{33} + 514819 p^{15} T^{34} - 10419 p^{16} T^{35} + 1029 p^{17} T^{36} - 11 p^{18} T^{37} + p^{19} T^{38} \) | |
| 97 | \( 1 - 36 T + 1255 T^{2} - 29968 T^{3} + 651236 T^{4} - 11947194 T^{5} + 201820377 T^{6} - 3068806610 T^{7} + 43790778370 T^{8} - 580665476778 T^{9} + 7361874661525 T^{10} - 88766645957284 T^{11} + 1041120566766923 T^{12} - 11821561759215710 T^{13} + 132070152530866706 T^{14} - 1440093805767598542 T^{15} + 15473149850567193298 T^{16} - \)\(16\!\cdots\!18\)\( T^{17} + \)\(16\!\cdots\!13\)\( T^{18} - \)\(16\!\cdots\!88\)\( T^{19} + \)\(16\!\cdots\!13\)\( p T^{20} - \)\(16\!\cdots\!18\)\( p^{2} T^{21} + 15473149850567193298 p^{3} T^{22} - 1440093805767598542 p^{4} T^{23} + 132070152530866706 p^{5} T^{24} - 11821561759215710 p^{6} T^{25} + 1041120566766923 p^{7} T^{26} - 88766645957284 p^{8} T^{27} + 7361874661525 p^{9} T^{28} - 580665476778 p^{10} T^{29} + 43790778370 p^{11} T^{30} - 3068806610 p^{12} T^{31} + 201820377 p^{13} T^{32} - 11947194 p^{14} T^{33} + 651236 p^{15} T^{34} - 29968 p^{16} T^{35} + 1255 p^{17} T^{36} - 36 p^{18} T^{37} + p^{19} T^{38} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{38} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.40163288442299338654884529300, −2.32156410186170697615815719694, −2.22431957232485353389539490034, −2.21557109395192254891682327495, −2.19095600289254623169227932908, −2.15736176422532099242698237274, −2.01210037766059153726528861229, −1.95040122112175422837680253533, −1.90706477126586889456208308937, −1.81784130788342957708718338168, −1.70141211825815803925597649775, −1.66201749664512394124435008946, −1.58290624371752045044603173505, −1.46402826057243351748832436356, −1.43418327362862452308148432475, −1.37629247716043539251167063925, −1.12656838359313271332800992460, −1.08053269392262110825486746873, −0.926507428058740749370233676131, −0.919983366968794242497869123708, −0.906358027405205879196940641477, −0.829129182955907352487188175723, −0.52215114753751240788379795000, −0.42936050096662434865186241315, −0.39902228955380175252813270664, 0.39902228955380175252813270664, 0.42936050096662434865186241315, 0.52215114753751240788379795000, 0.829129182955907352487188175723, 0.906358027405205879196940641477, 0.919983366968794242497869123708, 0.926507428058740749370233676131, 1.08053269392262110825486746873, 1.12656838359313271332800992460, 1.37629247716043539251167063925, 1.43418327362862452308148432475, 1.46402826057243351748832436356, 1.58290624371752045044603173505, 1.66201749664512394124435008946, 1.70141211825815803925597649775, 1.81784130788342957708718338168, 1.90706477126586889456208308937, 1.95040122112175422837680253533, 2.01210037766059153726528861229, 2.15736176422532099242698237274, 2.19095600289254623169227932908, 2.21557109395192254891682327495, 2.22431957232485353389539490034, 2.32156410186170697615815719694, 2.40163288442299338654884529300
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.