Dirichlet series
| L(s) = 1 | + 2-s − 2·3-s + 6·4-s − 8·5-s − 2·6-s + 6·7-s + 3·8-s + 4·9-s − 8·10-s + 9·11-s − 12·12-s − 8·13-s + 6·14-s + 16·15-s + 21·16-s + 12·17-s + 4·18-s − 22·19-s − 48·20-s − 12·21-s + 9·22-s + 3·23-s − 6·24-s + 28·25-s − 8·26-s + 2·27-s + 36·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 3·4-s − 3.57·5-s − 0.816·6-s + 2.26·7-s + 1.06·8-s + 4/3·9-s − 2.52·10-s + 2.71·11-s − 3.46·12-s − 2.21·13-s + 1.60·14-s + 4.13·15-s + 21/4·16-s + 2.91·17-s + 0.942·18-s − 5.04·19-s − 10.7·20-s − 2.61·21-s + 1.91·22-s + 0.625·23-s − 1.22·24-s + 28/5·25-s − 1.56·26-s + 0.384·27-s + 6.80·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 5^{16} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.13957\times 10^{10}\) |
| Root analytic conductor: | \(2.16130\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 5^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(6.384396391\) |
| \(L(\frac12)\) | \(\approx\) | \(6.384396391\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 2 T - 10 T^{3} - 8 T^{4} + 10 p T^{5} + 19 p T^{6} - 7 p^{2} T^{7} - 4 p^{4} T^{8} - 7 p^{3} T^{9} + 19 p^{3} T^{10} + 10 p^{4} T^{11} - 8 p^{4} T^{12} - 10 p^{5} T^{13} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 + T + T^{2} )^{8} \) | |
| 13 | \( ( 1 + T + T^{2} )^{8} \) | |
| good | 2 | \( 1 - T - 5 T^{2} + p^{3} T^{3} + p^{2} T^{4} - 5 p^{2} T^{5} + 19 T^{6} + 5 p T^{7} - 15 p T^{8} + 13 T^{9} - 27 T^{10} + 39 T^{11} + 51 T^{12} - 23 p^{2} T^{13} + 7 T^{14} + 19 T^{15} - T^{16} + 19 p T^{17} + 7 p^{2} T^{18} - 23 p^{5} T^{19} + 51 p^{4} T^{20} + 39 p^{5} T^{21} - 27 p^{6} T^{22} + 13 p^{7} T^{23} - 15 p^{9} T^{24} + 5 p^{10} T^{25} + 19 p^{10} T^{26} - 5 p^{13} T^{27} + p^{14} T^{28} + p^{16} T^{29} - 5 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) |
| 7 | \( 1 - 6 T - 18 T^{2} + 164 T^{3} + 213 T^{4} - 401 p T^{5} - 1374 T^{6} + 35450 T^{7} - 5249 T^{8} - 341477 T^{9} + 246473 T^{10} + 2584034 T^{11} - 3741641 T^{12} - 14140926 T^{13} + 5502762 p T^{14} + 37163016 T^{15} - 301992809 T^{16} + 37163016 p T^{17} + 5502762 p^{3} T^{18} - 14140926 p^{3} T^{19} - 3741641 p^{4} T^{20} + 2584034 p^{5} T^{21} + 246473 p^{6} T^{22} - 341477 p^{7} T^{23} - 5249 p^{8} T^{24} + 35450 p^{9} T^{25} - 1374 p^{10} T^{26} - 401 p^{12} T^{27} + 213 p^{12} T^{28} + 164 p^{13} T^{29} - 18 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 - 9 T - 5 T^{2} + 296 T^{3} - 494 T^{4} - 5342 T^{5} + 16153 T^{6} + 66670 T^{7} - 26484 p T^{8} - 714371 T^{9} + 4302202 T^{10} + 6946879 T^{11} - 58909531 T^{12} - 52730939 T^{13} + 740181953 T^{14} + 212131631 T^{15} - 8547104286 T^{16} + 212131631 p T^{17} + 740181953 p^{2} T^{18} - 52730939 p^{3} T^{19} - 58909531 p^{4} T^{20} + 6946879 p^{5} T^{21} + 4302202 p^{6} T^{22} - 714371 p^{7} T^{23} - 26484 p^{9} T^{24} + 66670 p^{9} T^{25} + 16153 p^{10} T^{26} - 5342 p^{11} T^{27} - 494 p^{12} T^{28} + 296 p^{13} T^{29} - 5 p^{14} T^{30} - 9 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( ( 1 - 6 T + 100 T^{2} - 555 T^{3} + 4886 T^{4} - 23913 T^{5} + 149072 T^{6} - 622493 T^{7} + 3060720 T^{8} - 622493 p T^{9} + 149072 p^{2} T^{10} - 23913 p^{3} T^{11} + 4886 p^{4} T^{12} - 555 p^{5} T^{13} + 100 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( ( 1 + 11 T + 131 T^{2} + 815 T^{3} + 5758 T^{4} + 27352 T^{5} + 162907 T^{6} + 36629 p T^{7} + 3621339 T^{8} + 36629 p^{2} T^{9} + 162907 p^{2} T^{10} + 27352 p^{3} T^{11} + 5758 p^{4} T^{12} + 815 p^{5} T^{13} + 131 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 3 T - 59 T^{2} + 538 T^{3} + 514 T^{4} - 23623 T^{5} + 104608 T^{6} + 311480 T^{7} - 4677660 T^{8} + 13214027 T^{9} + 73388287 T^{10} - 645899770 T^{11} + 693034868 T^{12} + 12500505935 T^{13} - 56441824915 T^{14} - 101270160710 T^{15} + 1536557670753 T^{16} - 101270160710 p T^{17} - 56441824915 p^{2} T^{18} + 12500505935 p^{3} T^{19} + 693034868 p^{4} T^{20} - 645899770 p^{5} T^{21} + 73388287 p^{6} T^{22} + 13214027 p^{7} T^{23} - 4677660 p^{8} T^{24} + 311480 p^{9} T^{25} + 104608 p^{10} T^{26} - 23623 p^{11} T^{27} + 514 p^{12} T^{28} + 538 p^{13} T^{29} - 59 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 - 8 T - 27 T^{2} + 700 T^{3} - 3670 T^{4} - 3885 T^{5} + 185163 T^{6} - 1149663 T^{7} + 1507216 T^{8} + 29591820 T^{9} - 246551863 T^{10} + 631951330 T^{11} + 3732918040 T^{12} - 38302084517 T^{13} + 110398712844 T^{14} + 505480022184 T^{15} - 5725084499690 T^{16} + 505480022184 p T^{17} + 110398712844 p^{2} T^{18} - 38302084517 p^{3} T^{19} + 3732918040 p^{4} T^{20} + 631951330 p^{5} T^{21} - 246551863 p^{6} T^{22} + 29591820 p^{7} T^{23} + 1507216 p^{8} T^{24} - 1149663 p^{9} T^{25} + 185163 p^{10} T^{26} - 3885 p^{11} T^{27} - 3670 p^{12} T^{28} + 700 p^{13} T^{29} - 27 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 18 T + 21 T^{2} + 1064 T^{3} - 206 T^{4} - 52970 T^{5} - 202008 T^{6} + 3053011 T^{7} + 13464505 T^{8} - 119491259 T^{9} - 751134713 T^{10} + 3842798386 T^{11} + 33375064407 T^{12} - 82955249767 T^{13} - 1339638492659 T^{14} + 665820246526 T^{15} + 48094755886364 T^{16} + 665820246526 p T^{17} - 1339638492659 p^{2} T^{18} - 82955249767 p^{3} T^{19} + 33375064407 p^{4} T^{20} + 3842798386 p^{5} T^{21} - 751134713 p^{6} T^{22} - 119491259 p^{7} T^{23} + 13464505 p^{8} T^{24} + 3053011 p^{9} T^{25} - 202008 p^{10} T^{26} - 52970 p^{11} T^{27} - 206 p^{12} T^{28} + 1064 p^{13} T^{29} + 21 p^{14} T^{30} - 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( ( 1 + 18 T + 235 T^{2} + 2251 T^{3} + 20009 T^{4} + 149618 T^{5} + 1081470 T^{6} + 7069204 T^{7} + 45538985 T^{8} + 7069204 p T^{9} + 1081470 p^{2} T^{10} + 149618 p^{3} T^{11} + 20009 p^{4} T^{12} + 2251 p^{5} T^{13} + 235 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 + 17 T - 40 T^{2} - 2007 T^{3} - 907 T^{4} + 130396 T^{5} + 14330 T^{6} - 6716679 T^{7} + 10264717 T^{8} + 295284466 T^{9} - 1209026009 T^{10} - 11321736870 T^{11} + 82159019872 T^{12} + 341358615032 T^{13} - 4267963983835 T^{14} - 5214819667413 T^{15} + 187372119611907 T^{16} - 5214819667413 p T^{17} - 4267963983835 p^{2} T^{18} + 341358615032 p^{3} T^{19} + 82159019872 p^{4} T^{20} - 11321736870 p^{5} T^{21} - 1209026009 p^{6} T^{22} + 295284466 p^{7} T^{23} + 10264717 p^{8} T^{24} - 6716679 p^{9} T^{25} + 14330 p^{10} T^{26} + 130396 p^{11} T^{27} - 907 p^{12} T^{28} - 2007 p^{13} T^{29} - 40 p^{14} T^{30} + 17 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 17 T - 58 T^{2} + 1941 T^{3} + 5363 T^{4} - 137152 T^{5} - 770524 T^{6} + 9147085 T^{7} + 62524475 T^{8} - 466233324 T^{9} - 4270045905 T^{10} + 19305066152 T^{11} + 255527055722 T^{12} - 613840898772 T^{13} - 13423077375831 T^{14} + 9007379983127 T^{15} + 627865126951657 T^{16} + 9007379983127 p T^{17} - 13423077375831 p^{2} T^{18} - 613840898772 p^{3} T^{19} + 255527055722 p^{4} T^{20} + 19305066152 p^{5} T^{21} - 4270045905 p^{6} T^{22} - 466233324 p^{7} T^{23} + 62524475 p^{8} T^{24} + 9147085 p^{9} T^{25} - 770524 p^{10} T^{26} - 137152 p^{11} T^{27} + 5363 p^{12} T^{28} + 1941 p^{13} T^{29} - 58 p^{14} T^{30} - 17 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 - 11 T - 158 T^{2} + 2319 T^{3} + 11388 T^{4} - 244569 T^{5} - 291560 T^{6} + 15624059 T^{7} - 14992833 T^{8} - 583763478 T^{9} + 1336015696 T^{10} + 6835966826 T^{11} + 10790104144 T^{12} + 427478003190 T^{13} - 6827521237386 T^{14} - 14368165872184 T^{15} + 473650701037160 T^{16} - 14368165872184 p T^{17} - 6827521237386 p^{2} T^{18} + 427478003190 p^{3} T^{19} + 10790104144 p^{4} T^{20} + 6835966826 p^{5} T^{21} + 1336015696 p^{6} T^{22} - 583763478 p^{7} T^{23} - 14992833 p^{8} T^{24} + 15624059 p^{9} T^{25} - 291560 p^{10} T^{26} - 244569 p^{11} T^{27} + 11388 p^{12} T^{28} + 2319 p^{13} T^{29} - 158 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( ( 1 - 10 T + 258 T^{2} - 1464 T^{3} + 25588 T^{4} - 77500 T^{5} + 1576849 T^{6} - 2262887 T^{7} + 84026442 T^{8} - 2262887 p T^{9} + 1576849 p^{2} T^{10} - 77500 p^{3} T^{11} + 25588 p^{4} T^{12} - 1464 p^{5} T^{13} + 258 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 - 7 T - 75 T^{2} + 46 T^{3} + 7688 T^{4} - 17140 T^{5} + 3033 p T^{6} - 3257722 T^{7} - 6679302 T^{8} - 78861133 T^{9} + 3098795926 T^{10} - 5526913739 T^{11} + 17327125317 T^{12} - 1112255898109 T^{13} + 2907608439611 T^{14} - 27103226651433 T^{15} + 610894242267270 T^{16} - 27103226651433 p T^{17} + 2907608439611 p^{2} T^{18} - 1112255898109 p^{3} T^{19} + 17327125317 p^{4} T^{20} - 5526913739 p^{5} T^{21} + 3098795926 p^{6} T^{22} - 78861133 p^{7} T^{23} - 6679302 p^{8} T^{24} - 3257722 p^{9} T^{25} + 3033 p^{11} T^{26} - 17140 p^{11} T^{27} + 7688 p^{12} T^{28} + 46 p^{13} T^{29} - 75 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 21 T + 21 T^{2} + 534 T^{3} + 26023 T^{4} - 168151 T^{5} - 1291244 T^{6} - 9315494 T^{7} + 156575888 T^{8} + 1051468901 T^{9} - 2295855458 T^{10} - 93743086867 T^{11} - 423742948435 T^{12} + 5765170705055 T^{13} + 27740342236662 T^{14} + 6611478498 T^{15} - 3478419960055630 T^{16} + 6611478498 p T^{17} + 27740342236662 p^{2} T^{18} + 5765170705055 p^{3} T^{19} - 423742948435 p^{4} T^{20} - 93743086867 p^{5} T^{21} - 2295855458 p^{6} T^{22} + 1051468901 p^{7} T^{23} + 156575888 p^{8} T^{24} - 9315494 p^{9} T^{25} - 1291244 p^{10} T^{26} - 168151 p^{11} T^{27} + 26023 p^{12} T^{28} + 534 p^{13} T^{29} + 21 p^{14} T^{30} - 21 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 13 T - 176 T^{2} + 4169 T^{3} - 1404 T^{4} - 516929 T^{5} + 2800619 T^{6} + 32136390 T^{7} - 320055097 T^{8} - 1166614855 T^{9} + 21299059523 T^{10} + 31302814701 T^{11} - 1196337275214 T^{12} - 55519830979 T^{13} + 895347650804 p T^{14} - 32890682620132 T^{15} - 3179900419606053 T^{16} - 32890682620132 p T^{17} + 895347650804 p^{3} T^{18} - 55519830979 p^{3} T^{19} - 1196337275214 p^{4} T^{20} + 31302814701 p^{5} T^{21} + 21299059523 p^{6} T^{22} - 1166614855 p^{7} T^{23} - 320055097 p^{8} T^{24} + 32136390 p^{9} T^{25} + 2800619 p^{10} T^{26} - 516929 p^{11} T^{27} - 1404 p^{12} T^{28} + 4169 p^{13} T^{29} - 176 p^{14} T^{30} - 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 34 T + 865 T^{2} + 15373 T^{3} + 232729 T^{4} + 2907377 T^{5} + 32555353 T^{6} + 317949849 T^{7} + 2846223712 T^{8} + 317949849 p T^{9} + 32555353 p^{2} T^{10} + 2907377 p^{3} T^{11} + 232729 p^{4} T^{12} + 15373 p^{5} T^{13} + 865 p^{6} T^{14} + 34 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 16 T + 388 T^{2} + 4789 T^{3} + 69692 T^{4} + 722871 T^{5} + 8152536 T^{6} + 73194077 T^{7} + 692947628 T^{8} + 73194077 p T^{9} + 8152536 p^{2} T^{10} + 722871 p^{3} T^{11} + 69692 p^{4} T^{12} + 4789 p^{5} T^{13} + 388 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 - 37 T + 321 T^{2} + 3544 T^{3} - 40986 T^{4} - 822165 T^{5} + 11337251 T^{6} + 49818407 T^{7} - 1120137888 T^{8} - 4947701274 T^{9} + 121478417398 T^{10} + 171914446773 T^{11} - 9182230819820 T^{12} - 3410637537796 T^{13} + 620941780818050 T^{14} + 339385919908394 T^{15} - 50384385030944825 T^{16} + 339385919908394 p T^{17} + 620941780818050 p^{2} T^{18} - 3410637537796 p^{3} T^{19} - 9182230819820 p^{4} T^{20} + 171914446773 p^{5} T^{21} + 121478417398 p^{6} T^{22} - 4947701274 p^{7} T^{23} - 1120137888 p^{8} T^{24} + 49818407 p^{9} T^{25} + 11337251 p^{10} T^{26} - 822165 p^{11} T^{27} - 40986 p^{12} T^{28} + 3544 p^{13} T^{29} + 321 p^{14} T^{30} - 37 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 3 T - 195 T^{2} + 2812 T^{3} + 9352 T^{4} - 482317 T^{5} + 3210177 T^{6} + 34579853 T^{7} - 620889458 T^{8} + 1286702090 T^{9} + 56054373982 T^{10} - 506034278639 T^{11} - 13369040184 p T^{12} + 56824716823148 T^{13} - 330516278305084 T^{14} - 2154204196638402 T^{15} + 46906329253721967 T^{16} - 2154204196638402 p T^{17} - 330516278305084 p^{2} T^{18} + 56824716823148 p^{3} T^{19} - 13369040184 p^{5} T^{20} - 506034278639 p^{5} T^{21} + 56054373982 p^{6} T^{22} + 1286702090 p^{7} T^{23} - 620889458 p^{8} T^{24} + 34579853 p^{9} T^{25} + 3210177 p^{10} T^{26} - 482317 p^{11} T^{27} + 9352 p^{12} T^{28} + 2812 p^{13} T^{29} - 195 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 14 T + 616 T^{2} - 5933 T^{3} + 153805 T^{4} - 1078132 T^{5} + 22379338 T^{6} - 122929612 T^{7} + 2291032834 T^{8} - 122929612 p T^{9} + 22379338 p^{2} T^{10} - 1078132 p^{3} T^{11} + 153805 p^{4} T^{12} - 5933 p^{5} T^{13} + 616 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 17 T - 379 T^{2} + 5904 T^{3} + 106330 T^{4} - 1210550 T^{5} - 22087776 T^{6} + 167115511 T^{7} + 3610588903 T^{8} - 16322112125 T^{9} - 493231660965 T^{10} + 1186570786416 T^{11} + 58528296153207 T^{12} - 63016748892956 T^{13} - 6294345209098497 T^{14} + 1693731077083404 T^{15} + 632631145452769018 T^{16} + 1693731077083404 p T^{17} - 6294345209098497 p^{2} T^{18} - 63016748892956 p^{3} T^{19} + 58528296153207 p^{4} T^{20} + 1186570786416 p^{5} T^{21} - 493231660965 p^{6} T^{22} - 16322112125 p^{7} T^{23} + 3610588903 p^{8} T^{24} + 167115511 p^{9} T^{25} - 22087776 p^{10} T^{26} - 1210550 p^{11} T^{27} + 106330 p^{12} T^{28} + 5904 p^{13} T^{29} - 379 p^{14} T^{30} - 17 p^{15} T^{31} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.84842390999323525794732735798, −2.74728019912266350864117378003, −2.73033770458486384152356992801, −2.59270251401363407547948165554, −2.49899915290145620932097608246, −2.34380128458081775240095390717, −2.32688347094307509248280764551, −2.31257954848822374243134253816, −2.27271791927242523388127769691, −2.11183885807659092785298238530, −2.03147740279764700657821500032, −1.97474153191801544114127374079, −1.89934266470487007404717207214, −1.67403722022960257432981975462, −1.66027798782257428105105098378, −1.46247097402448756162657487871, −1.33640235581003778042805293805, −1.29902081219698316102674225042, −1.15449808273147756373973414148, −0.852743883679145093490079907122, −0.78246052598545691982104794144, −0.75113314085972478079123530874, −0.72289600615800499607600480988, −0.64757881202566143244974146085, −0.12009540566764818605671167025, 0.12009540566764818605671167025, 0.64757881202566143244974146085, 0.72289600615800499607600480988, 0.75113314085972478079123530874, 0.78246052598545691982104794144, 0.852743883679145093490079907122, 1.15449808273147756373973414148, 1.29902081219698316102674225042, 1.33640235581003778042805293805, 1.46247097402448756162657487871, 1.66027798782257428105105098378, 1.67403722022960257432981975462, 1.89934266470487007404717207214, 1.97474153191801544114127374079, 2.03147740279764700657821500032, 2.11183885807659092785298238530, 2.27271791927242523388127769691, 2.31257954848822374243134253816, 2.32688347094307509248280764551, 2.34380128458081775240095390717, 2.49899915290145620932097608246, 2.59270251401363407547948165554, 2.73033770458486384152356992801, 2.74728019912266350864117378003, 2.84842390999323525794732735798
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.