Dirichlet series
| L(s) = 1 | + 7·5-s − 2·7-s + 8·9-s − 10·11-s − 2·13-s − 5·17-s − 8·19-s + 8·23-s + 21·25-s − 6·27-s − 4·29-s + 16·31-s − 14·35-s + 4·37-s + 9·41-s + 22·43-s + 56·45-s + 36·47-s + 23·49-s + 2·53-s − 70·55-s + 12·59-s − 20·61-s − 16·63-s − 14·65-s + 12·67-s − 13·71-s + ⋯ |
| L(s) = 1 | + 3.13·5-s − 0.755·7-s + 8/3·9-s − 3.01·11-s − 0.554·13-s − 1.21·17-s − 1.83·19-s + 1.66·23-s + 21/5·25-s − 1.15·27-s − 0.742·29-s + 2.87·31-s − 2.36·35-s + 0.657·37-s + 1.40·41-s + 3.35·43-s + 8.34·45-s + 5.25·47-s + 23/7·49-s + 0.274·53-s − 9.43·55-s + 1.56·59-s − 2.56·61-s − 2.01·63-s − 1.73·65-s + 1.46·67-s − 1.54·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{28} \cdot 5^{14} \cdot 37^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.32591\times 10^{10}\) |
| Root analytic conductor: | \(2.43082\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{28} \cdot 5^{14} \cdot 37^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(7.774651167\) |
| \(L(\frac12)\) | \(\approx\) | \(7.774651167\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - T + T^{2} )^{7} \) | |
| 37 | \( 1 - 4 T - 59 T^{2} - 138 T^{3} + 63 p T^{4} + 21864 T^{5} - 52970 T^{6} - 873019 T^{7} - 52970 p T^{8} + 21864 p^{2} T^{9} + 63 p^{4} T^{10} - 138 p^{4} T^{11} - 59 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| good | 3 | \( 1 - 8 T^{2} + 2 p T^{3} + 31 T^{4} - 55 T^{5} - 53 T^{6} + 256 T^{7} - 115 T^{8} - 329 p T^{9} + 1162 T^{10} + 2791 T^{11} - 5735 T^{12} - 383 p^{2} T^{13} + 6989 p T^{14} - 383 p^{3} T^{15} - 5735 p^{2} T^{16} + 2791 p^{3} T^{17} + 1162 p^{4} T^{18} - 329 p^{6} T^{19} - 115 p^{6} T^{20} + 256 p^{7} T^{21} - 53 p^{8} T^{22} - 55 p^{9} T^{23} + 31 p^{10} T^{24} + 2 p^{12} T^{25} - 8 p^{12} T^{26} + p^{14} T^{28} \) |
| 7 | \( 1 + 2 T - 19 T^{2} - 86 T^{3} + 73 T^{4} + 1094 T^{5} + 1956 T^{6} - 4835 T^{7} - 26128 T^{8} - 40661 T^{9} + 88598 T^{10} + 630758 T^{11} + 1191406 T^{12} - 2469610 T^{13} - 14934709 T^{14} - 2469610 p T^{15} + 1191406 p^{2} T^{16} + 630758 p^{3} T^{17} + 88598 p^{4} T^{18} - 40661 p^{5} T^{19} - 26128 p^{6} T^{20} - 4835 p^{7} T^{21} + 1956 p^{8} T^{22} + 1094 p^{9} T^{23} + 73 p^{10} T^{24} - 86 p^{11} T^{25} - 19 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 11 | \( ( 1 + 5 T + 28 T^{2} + 78 T^{3} + 358 T^{4} + 1266 T^{5} + 6279 T^{6} + 20189 T^{7} + 6279 p T^{8} + 1266 p^{2} T^{9} + 358 p^{3} T^{10} + 78 p^{4} T^{11} + 28 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 13 | \( 1 + 2 T - 41 T^{2} - 118 T^{3} + 527 T^{4} + 2079 T^{5} - 4998 T^{6} - 6521 T^{7} + 193798 T^{8} + 29329 T^{9} - 4388431 T^{10} - 5373734 T^{11} + 45249114 T^{12} + 53995795 T^{13} - 380184687 T^{14} + 53995795 p T^{15} + 45249114 p^{2} T^{16} - 5373734 p^{3} T^{17} - 4388431 p^{4} T^{18} + 29329 p^{5} T^{19} + 193798 p^{6} T^{20} - 6521 p^{7} T^{21} - 4998 p^{8} T^{22} + 2079 p^{9} T^{23} + 527 p^{10} T^{24} - 118 p^{11} T^{25} - 41 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 17 | \( 1 + 5 T - 41 T^{2} - 220 T^{3} + 710 T^{4} + 3976 T^{5} - 8913 T^{6} - 3318 p T^{7} + 84389 T^{8} + 934504 T^{9} - 418550 T^{10} - 10472379 T^{11} + 24970193 T^{12} + 44914042 T^{13} - 794403549 T^{14} + 44914042 p T^{15} + 24970193 p^{2} T^{16} - 10472379 p^{3} T^{17} - 418550 p^{4} T^{18} + 934504 p^{5} T^{19} + 84389 p^{6} T^{20} - 3318 p^{8} T^{21} - 8913 p^{8} T^{22} + 3976 p^{9} T^{23} + 710 p^{10} T^{24} - 220 p^{11} T^{25} - 41 p^{12} T^{26} + 5 p^{13} T^{27} + p^{14} T^{28} \) | |
| 19 | \( 1 + 8 T + 7 T^{2} + 86 T^{3} + 1142 T^{4} + 891 T^{5} + 6192 T^{6} + 119734 T^{7} - 21965 T^{8} - 982553 T^{9} + 2283827 T^{10} - 20940446 T^{11} - 141852513 T^{12} - 417913172 T^{13} - 2567303703 T^{14} - 417913172 p T^{15} - 141852513 p^{2} T^{16} - 20940446 p^{3} T^{17} + 2283827 p^{4} T^{18} - 982553 p^{5} T^{19} - 21965 p^{6} T^{20} + 119734 p^{7} T^{21} + 6192 p^{8} T^{22} + 891 p^{9} T^{23} + 1142 p^{10} T^{24} + 86 p^{11} T^{25} + 7 p^{12} T^{26} + 8 p^{13} T^{27} + p^{14} T^{28} \) | |
| 23 | \( ( 1 - 4 T + 86 T^{2} - 392 T^{3} + 4116 T^{4} - 16957 T^{5} + 136474 T^{6} - 465295 T^{7} + 136474 p T^{8} - 16957 p^{2} T^{9} + 4116 p^{3} T^{10} - 392 p^{4} T^{11} + 86 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 29 | \( ( 1 + 2 T + 91 T^{2} + 254 T^{3} + 187 p T^{4} + 14146 T^{5} + 214500 T^{6} + 523439 T^{7} + 214500 p T^{8} + 14146 p^{2} T^{9} + 187 p^{4} T^{10} + 254 p^{4} T^{11} + 91 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 31 | \( ( 1 - 8 T + 148 T^{2} - 782 T^{3} + 9403 T^{4} - 37238 T^{5} + 377519 T^{6} - 1242853 T^{7} + 377519 p T^{8} - 37238 p^{2} T^{9} + 9403 p^{3} T^{10} - 782 p^{4} T^{11} + 148 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 41 | \( 1 - 9 T - 40 T^{2} - 773 T^{3} + 10721 T^{4} + 40533 T^{5} + 216455 T^{6} - 5588498 T^{7} - 20501062 T^{8} - 23785140 T^{9} + 1605898872 T^{10} + 7186406688 T^{11} + 9736207456 T^{12} - 246387070821 T^{13} - 2034326429285 T^{14} - 246387070821 p T^{15} + 9736207456 p^{2} T^{16} + 7186406688 p^{3} T^{17} + 1605898872 p^{4} T^{18} - 23785140 p^{5} T^{19} - 20501062 p^{6} T^{20} - 5588498 p^{7} T^{21} + 216455 p^{8} T^{22} + 40533 p^{9} T^{23} + 10721 p^{10} T^{24} - 773 p^{11} T^{25} - 40 p^{12} T^{26} - 9 p^{13} T^{27} + p^{14} T^{28} \) | |
| 43 | \( ( 1 - 11 T + 6 p T^{2} - 2177 T^{3} + 29974 T^{4} - 202899 T^{5} + 2035831 T^{6} - 11096581 T^{7} + 2035831 p T^{8} - 202899 p^{2} T^{9} + 29974 p^{3} T^{10} - 2177 p^{4} T^{11} + 6 p^{6} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 47 | \( ( 1 - 18 T + 310 T^{2} - 3367 T^{3} + 36537 T^{4} - 312245 T^{5} + 2640261 T^{6} - 18337209 T^{7} + 2640261 p T^{8} - 312245 p^{2} T^{9} + 36537 p^{3} T^{10} - 3367 p^{4} T^{11} + 310 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 53 | \( 1 - 2 T - 3 p T^{2} + 776 T^{3} + 13295 T^{4} - 89180 T^{5} - 393977 T^{6} + 4973959 T^{7} - 17227541 T^{8} - 50662459 T^{9} + 2322825614 T^{10} - 9150700548 T^{11} - 112140022419 T^{12} + 317261384462 T^{13} + 4570136006101 T^{14} + 317261384462 p T^{15} - 112140022419 p^{2} T^{16} - 9150700548 p^{3} T^{17} + 2322825614 p^{4} T^{18} - 50662459 p^{5} T^{19} - 17227541 p^{6} T^{20} + 4973959 p^{7} T^{21} - 393977 p^{8} T^{22} - 89180 p^{9} T^{23} + 13295 p^{10} T^{24} + 776 p^{11} T^{25} - 3 p^{13} T^{26} - 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 59 | \( 1 - 12 T - 137 T^{2} - 262 T^{3} + 40098 T^{4} + 80315 T^{5} - 2629760 T^{6} - 41347040 T^{7} + 184397527 T^{8} + 3448894033 T^{9} + 14924754993 T^{10} - 287419164774 T^{11} - 1784754769889 T^{12} + 4207885000658 T^{13} + 181519537832993 T^{14} + 4207885000658 p T^{15} - 1784754769889 p^{2} T^{16} - 287419164774 p^{3} T^{17} + 14924754993 p^{4} T^{18} + 3448894033 p^{5} T^{19} + 184397527 p^{6} T^{20} - 41347040 p^{7} T^{21} - 2629760 p^{8} T^{22} + 80315 p^{9} T^{23} + 40098 p^{10} T^{24} - 262 p^{11} T^{25} - 137 p^{12} T^{26} - 12 p^{13} T^{27} + p^{14} T^{28} \) | |
| 61 | \( 1 + 20 T + 79 T^{2} + 1322 T^{3} + 37930 T^{4} + 177119 T^{5} + 216780 T^{6} + 32861240 T^{7} + 246213301 T^{8} - 360988199 T^{9} + 16431774185 T^{10} + 230488056862 T^{11} - 26758298599 T^{12} + 4276850293946 T^{13} + 138514122546815 T^{14} + 4276850293946 p T^{15} - 26758298599 p^{2} T^{16} + 230488056862 p^{3} T^{17} + 16431774185 p^{4} T^{18} - 360988199 p^{5} T^{19} + 246213301 p^{6} T^{20} + 32861240 p^{7} T^{21} + 216780 p^{8} T^{22} + 177119 p^{9} T^{23} + 37930 p^{10} T^{24} + 1322 p^{11} T^{25} + 79 p^{12} T^{26} + 20 p^{13} T^{27} + p^{14} T^{28} \) | |
| 67 | \( 1 - 12 T - 168 T^{2} + 24 p T^{3} + 19663 T^{4} - 63026 T^{5} - 2286210 T^{6} + 1431055 T^{7} + 194703388 T^{8} - 139510770 T^{9} - 12296901922 T^{10} - 3059738788 T^{11} + 855093360770 T^{12} + 520202986907 T^{13} - 62915756488475 T^{14} + 520202986907 p T^{15} + 855093360770 p^{2} T^{16} - 3059738788 p^{3} T^{17} - 12296901922 p^{4} T^{18} - 139510770 p^{5} T^{19} + 194703388 p^{6} T^{20} + 1431055 p^{7} T^{21} - 2286210 p^{8} T^{22} - 63026 p^{9} T^{23} + 19663 p^{10} T^{24} + 24 p^{12} T^{25} - 168 p^{12} T^{26} - 12 p^{13} T^{27} + p^{14} T^{28} \) | |
| 71 | \( 1 + 13 T + 106 T^{2} - 35 p T^{3} - 41709 T^{4} - 399241 T^{5} + 2419285 T^{6} + 62328488 T^{7} + 698184996 T^{8} - 377577578 T^{9} - 55316032620 T^{10} - 748798407582 T^{11} - 1661384372702 T^{12} + 30085198291995 T^{13} + 539795012991877 T^{14} + 30085198291995 p T^{15} - 1661384372702 p^{2} T^{16} - 748798407582 p^{3} T^{17} - 55316032620 p^{4} T^{18} - 377577578 p^{5} T^{19} + 698184996 p^{6} T^{20} + 62328488 p^{7} T^{21} + 2419285 p^{8} T^{22} - 399241 p^{9} T^{23} - 41709 p^{10} T^{24} - 35 p^{12} T^{25} + 106 p^{12} T^{26} + 13 p^{13} T^{27} + p^{14} T^{28} \) | |
| 73 | \( ( 1 + 6 T + 231 T^{2} + 1770 T^{3} + 26930 T^{4} + 228818 T^{5} + 2221482 T^{6} + 19524575 T^{7} + 2221482 p T^{8} + 228818 p^{2} T^{9} + 26930 p^{3} T^{10} + 1770 p^{4} T^{11} + 231 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 79 | \( 1 + 21 T - 29 T^{2} - 3980 T^{3} - 15465 T^{4} + 446087 T^{5} + 4675858 T^{6} - 9191886 T^{7} - 461842622 T^{8} - 2633630929 T^{9} + 21075323817 T^{10} + 374237280415 T^{11} + 1223295287327 T^{12} - 13818126107014 T^{13} - 195780554887585 T^{14} - 13818126107014 p T^{15} + 1223295287327 p^{2} T^{16} + 374237280415 p^{3} T^{17} + 21075323817 p^{4} T^{18} - 2633630929 p^{5} T^{19} - 461842622 p^{6} T^{20} - 9191886 p^{7} T^{21} + 4675858 p^{8} T^{22} + 446087 p^{9} T^{23} - 15465 p^{10} T^{24} - 3980 p^{11} T^{25} - 29 p^{12} T^{26} + 21 p^{13} T^{27} + p^{14} T^{28} \) | |
| 83 | \( 1 - 7 T - 287 T^{2} + 4036 T^{3} + 35580 T^{4} - 933656 T^{5} - 664322 T^{6} + 137279302 T^{7} - 549288219 T^{8} - 14064302623 T^{9} + 121285087326 T^{10} + 978466018056 T^{11} - 15694580904839 T^{12} - 31617676335006 T^{13} + 1487646551439841 T^{14} - 31617676335006 p T^{15} - 15694580904839 p^{2} T^{16} + 978466018056 p^{3} T^{17} + 121285087326 p^{4} T^{18} - 14064302623 p^{5} T^{19} - 549288219 p^{6} T^{20} + 137279302 p^{7} T^{21} - 664322 p^{8} T^{22} - 933656 p^{9} T^{23} + 35580 p^{10} T^{24} + 4036 p^{11} T^{25} - 287 p^{12} T^{26} - 7 p^{13} T^{27} + p^{14} T^{28} \) | |
| 89 | \( 1 - 16 T - 205 T^{2} + 3376 T^{3} + 37780 T^{4} - 457458 T^{5} - 4169773 T^{6} + 27122665 T^{7} + 374369293 T^{8} - 309919429 T^{9} - 12517763791 T^{10} - 164329762752 T^{11} - 461147298123 T^{12} + 6149948062296 T^{13} + 150963647111625 T^{14} + 6149948062296 p T^{15} - 461147298123 p^{2} T^{16} - 164329762752 p^{3} T^{17} - 12517763791 p^{4} T^{18} - 309919429 p^{5} T^{19} + 374369293 p^{6} T^{20} + 27122665 p^{7} T^{21} - 4169773 p^{8} T^{22} - 457458 p^{9} T^{23} + 37780 p^{10} T^{24} + 3376 p^{11} T^{25} - 205 p^{12} T^{26} - 16 p^{13} T^{27} + p^{14} T^{28} \) | |
| 97 | \( ( 1 + 3 T + 312 T^{2} + 2424 T^{3} + 51250 T^{4} + 528674 T^{5} + 6697797 T^{6} + 62462713 T^{7} + 6697797 p T^{8} + 528674 p^{2} T^{9} + 51250 p^{3} T^{10} + 2424 p^{4} T^{11} + 312 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.93896879235750388085778278123, −2.80099158824427797775372476648, −2.68925904593835137500670089490, −2.65113198834744776134413210401, −2.64752936723124376683221236787, −2.51364142827945246154785164941, −2.49095445957088181066527333000, −2.46014635408893390709856844810, −2.21993078792121678521439973179, −2.15783323766442614333595535102, −2.13779129268000012119736149247, −2.05171653624446001398053543600, −1.99946953772921899809399869473, −1.86244546990676808904428669072, −1.83847204381232648412976142556, −1.63229463532993444019638277153, −1.24802014668690431906539345488, −1.23722084599946028653655376445, −1.12243559075917509180469464197, −1.05590450437346552391641636914, −1.00000918527383678780213226869, −0.911919968132864920701668004454, −0.73310837383364321488684089190, −0.27459153328579707107217081653, −0.20348910625913617407767134571, 0.20348910625913617407767134571, 0.27459153328579707107217081653, 0.73310837383364321488684089190, 0.911919968132864920701668004454, 1.00000918527383678780213226869, 1.05590450437346552391641636914, 1.12243559075917509180469464197, 1.23722084599946028653655376445, 1.24802014668690431906539345488, 1.63229463532993444019638277153, 1.83847204381232648412976142556, 1.86244546990676808904428669072, 1.99946953772921899809399869473, 2.05171653624446001398053543600, 2.13779129268000012119736149247, 2.15783323766442614333595535102, 2.21993078792121678521439973179, 2.46014635408893390709856844810, 2.49095445957088181066527333000, 2.51364142827945246154785164941, 2.64752936723124376683221236787, 2.65113198834744776134413210401, 2.68925904593835137500670089490, 2.80099158824427797775372476648, 2.93896879235750388085778278123
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.