Dirichlet series
| L(s) = 1 | − 2·3-s + 18·4-s − 12·7-s − 9·9-s − 36·12-s + 32·13-s + 145·16-s − 24·19-s + 24·21-s + 168·25-s − 2·27-s − 216·28-s + 24·31-s − 162·36-s − 40·37-s − 64·39-s − 36·43-s − 290·48-s − 274·49-s + 576·52-s + 48·57-s − 8·61-s + 108·63-s + 731·64-s + 136·67-s − 68·73-s − 336·75-s + ⋯ |
| L(s) = 1 | − 2/3·3-s + 9/2·4-s − 1.71·7-s − 9-s − 3·12-s + 2.46·13-s + 9.06·16-s − 1.26·19-s + 8/7·21-s + 6.71·25-s − 0.0740·27-s − 7.71·28-s + 0.774·31-s − 9/2·36-s − 1.08·37-s − 1.64·39-s − 0.837·43-s − 6.04·48-s − 5.59·49-s + 11.0·52-s + 0.842·57-s − 0.131·61-s + 12/7·63-s + 11.4·64-s + 2.02·67-s − 0.931·73-s − 4.47·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s+1)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(3^{18} \cdot 29^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.58955\times 10^{6}\) |
| Root analytic conductor: | \(1.53966\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 3^{18} \cdot 29^{18} ,\ ( \ : [1]^{18} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.3010758505\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3010758505\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 2 T + 13 T^{2} + 46 T^{3} + 37 T^{4} + 328 T^{5} - 233 p T^{6} - 3598 T^{7} - 1096 p^{2} T^{8} - 2428 p^{3} T^{9} - 1096 p^{4} T^{10} - 3598 p^{4} T^{11} - 233 p^{7} T^{12} + 328 p^{8} T^{13} + 37 p^{10} T^{14} + 46 p^{12} T^{15} + 13 p^{14} T^{16} + 2 p^{16} T^{17} + p^{18} T^{18} \) |
| 29 | \( ( 1 + p T^{2} )^{9} \) | |
| good | 2 | \( 1 - 9 p T^{2} + 179 T^{4} - 1343 T^{6} + 8417 T^{8} - 46715 T^{10} + 238509 T^{12} - 1128161 T^{14} + 2486695 p T^{16} - 20548323 T^{18} + 2486695 p^{5} T^{20} - 1128161 p^{8} T^{22} + 238509 p^{12} T^{24} - 46715 p^{16} T^{26} + 8417 p^{20} T^{28} - 1343 p^{24} T^{30} + 179 p^{28} T^{32} - 9 p^{33} T^{34} + p^{36} T^{36} \) |
| 5 | \( 1 - 168 T^{2} + 15448 T^{4} - 203698 p T^{6} + 53300024 T^{8} - 2333771144 T^{10} + 88158227327 T^{12} - 585305986192 p T^{14} + 86364325403376 T^{16} - 2280293042501164 T^{18} + 86364325403376 p^{4} T^{20} - 585305986192 p^{9} T^{22} + 88158227327 p^{12} T^{24} - 2333771144 p^{16} T^{26} + 53300024 p^{20} T^{28} - 203698 p^{25} T^{30} + 15448 p^{28} T^{32} - 168 p^{32} T^{34} + p^{36} T^{36} \) | |
| 7 | \( ( 1 + 6 T + 191 T^{2} + 968 T^{3} + 16263 T^{4} + 6686 p T^{5} + 722651 T^{6} - 562944 T^{7} + 17600738 T^{8} - 120548504 T^{9} + 17600738 p^{2} T^{10} - 562944 p^{4} T^{11} + 722651 p^{6} T^{12} + 6686 p^{9} T^{13} + 16263 p^{10} T^{14} + 968 p^{12} T^{15} + 191 p^{14} T^{16} + 6 p^{16} T^{17} + p^{18} T^{18} )^{2} \) | |
| 11 | \( 1 - 1480 T^{2} + 1090740 T^{4} - 529820306 T^{6} + 189603627808 T^{8} - 52993750515760 T^{10} + 11974420199718087 T^{12} - 203164898702079200 p T^{14} + \)\(34\!\cdots\!88\)\( T^{16} - \)\(45\!\cdots\!36\)\( T^{18} + \)\(34\!\cdots\!88\)\( p^{4} T^{20} - 203164898702079200 p^{9} T^{22} + 11974420199718087 p^{12} T^{24} - 52993750515760 p^{16} T^{26} + 189603627808 p^{20} T^{28} - 529820306 p^{24} T^{30} + 1090740 p^{28} T^{32} - 1480 p^{32} T^{34} + p^{36} T^{36} \) | |
| 13 | \( ( 1 - 16 T + 752 T^{2} - 9502 T^{3} + 274068 T^{4} - 2799288 T^{5} + 67248371 T^{6} - 581232000 T^{7} + 13043602740 T^{8} - 102628851788 T^{9} + 13043602740 p^{2} T^{10} - 581232000 p^{4} T^{11} + 67248371 p^{6} T^{12} - 2799288 p^{8} T^{13} + 274068 p^{10} T^{14} - 9502 p^{12} T^{15} + 752 p^{14} T^{16} - 16 p^{16} T^{17} + p^{18} T^{18} )^{2} \) | |
| 17 | \( 1 - 3154 T^{2} + 4965359 T^{4} - 5187115040 T^{6} + 4033556125391 T^{8} - 2482367046821102 T^{10} + 1254543078205077319 T^{12} - \)\(53\!\cdots\!24\)\( T^{14} + \)\(19\!\cdots\!14\)\( T^{16} - \)\(60\!\cdots\!08\)\( T^{18} + \)\(19\!\cdots\!14\)\( p^{4} T^{20} - \)\(53\!\cdots\!24\)\( p^{8} T^{22} + 1254543078205077319 p^{12} T^{24} - 2482367046821102 p^{16} T^{26} + 4033556125391 p^{20} T^{28} - 5187115040 p^{24} T^{30} + 4965359 p^{28} T^{32} - 3154 p^{32} T^{34} + p^{36} T^{36} \) | |
| 19 | \( ( 1 + 12 T + 1772 T^{2} + 15324 T^{3} + 1392509 T^{4} + 9173888 T^{5} + 713592315 T^{6} + 4266359396 T^{7} + 296110515331 T^{8} + 1723465936104 T^{9} + 296110515331 p^{2} T^{10} + 4266359396 p^{4} T^{11} + 713592315 p^{6} T^{12} + 9173888 p^{8} T^{13} + 1392509 p^{10} T^{14} + 15324 p^{12} T^{15} + 1772 p^{14} T^{16} + 12 p^{16} T^{17} + p^{18} T^{18} )^{2} \) | |
| 23 | \( 1 - 4258 T^{2} + 8966049 T^{4} - 12109011968 T^{6} + 11388695607932 T^{8} - 7487887352268920 T^{10} + 3061497553021159676 T^{12} - \)\(20\!\cdots\!44\)\( T^{14} - \)\(73\!\cdots\!22\)\( T^{16} + \)\(58\!\cdots\!28\)\( T^{18} - \)\(73\!\cdots\!22\)\( p^{4} T^{20} - \)\(20\!\cdots\!44\)\( p^{8} T^{22} + 3061497553021159676 p^{12} T^{24} - 7487887352268920 p^{16} T^{26} + 11388695607932 p^{20} T^{28} - 12109011968 p^{24} T^{30} + 8966049 p^{28} T^{32} - 4258 p^{32} T^{34} + p^{36} T^{36} \) | |
| 31 | \( ( 1 - 12 T + 6121 T^{2} - 35542 T^{3} + 17417409 T^{4} - 30770114 T^{5} + 31687297605 T^{6} + 4668552798 T^{7} + 41280721770976 T^{8} + 23188101704876 T^{9} + 41280721770976 p^{2} T^{10} + 4668552798 p^{4} T^{11} + 31687297605 p^{6} T^{12} - 30770114 p^{8} T^{13} + 17417409 p^{10} T^{14} - 35542 p^{12} T^{15} + 6121 p^{14} T^{16} - 12 p^{16} T^{17} + p^{18} T^{18} )^{2} \) | |
| 37 | \( ( 1 + 20 T + 6752 T^{2} + 167040 T^{3} + 25201429 T^{4} + 611672640 T^{5} + 63434157823 T^{6} + 1433158974080 T^{7} + 115917415625171 T^{8} + 2324771025321560 T^{9} + 115917415625171 p^{2} T^{10} + 1433158974080 p^{4} T^{11} + 63434157823 p^{6} T^{12} + 611672640 p^{8} T^{13} + 25201429 p^{10} T^{14} + 167040 p^{12} T^{15} + 6752 p^{14} T^{16} + 20 p^{16} T^{17} + p^{18} T^{18} )^{2} \) | |
| 41 | \( 1 - 11892 T^{2} + 74887706 T^{4} - 328689404242 T^{6} + 1120586922985703 T^{8} - 3150621324959316660 T^{10} + \)\(75\!\cdots\!09\)\( T^{12} - \)\(16\!\cdots\!38\)\( T^{14} + \)\(75\!\cdots\!57\)\( p T^{16} - \)\(54\!\cdots\!08\)\( T^{18} + \)\(75\!\cdots\!57\)\( p^{5} T^{20} - \)\(16\!\cdots\!38\)\( p^{8} T^{22} + \)\(75\!\cdots\!09\)\( p^{12} T^{24} - 3150621324959316660 p^{16} T^{26} + 1120586922985703 p^{20} T^{28} - 328689404242 p^{24} T^{30} + 74887706 p^{28} T^{32} - 11892 p^{32} T^{34} + p^{36} T^{36} \) | |
| 43 | \( ( 1 + 18 T + 5688 T^{2} - 41020 T^{3} + 17783702 T^{4} - 413238386 T^{5} + 1048081941 p T^{6} - 1513398007008 T^{7} + 94684389334634 T^{8} - 3404989344914648 T^{9} + 94684389334634 p^{2} T^{10} - 1513398007008 p^{4} T^{11} + 1048081941 p^{7} T^{12} - 413238386 p^{8} T^{13} + 17783702 p^{10} T^{14} - 41020 p^{12} T^{15} + 5688 p^{14} T^{16} + 18 p^{16} T^{17} + p^{18} T^{18} )^{2} \) | |
| 47 | \( 1 - 24006 T^{2} + 290454537 T^{4} - 2342752939544 T^{6} + 14084117467998162 T^{8} - 66960045776254093044 T^{10} + \)\(26\!\cdots\!54\)\( T^{12} - \)\(85\!\cdots\!40\)\( T^{14} + \)\(23\!\cdots\!55\)\( T^{16} - \)\(56\!\cdots\!30\)\( T^{18} + \)\(23\!\cdots\!55\)\( p^{4} T^{20} - \)\(85\!\cdots\!40\)\( p^{8} T^{22} + \)\(26\!\cdots\!54\)\( p^{12} T^{24} - 66960045776254093044 p^{16} T^{26} + 14084117467998162 p^{20} T^{28} - 2342752939544 p^{24} T^{30} + 290454537 p^{28} T^{32} - 24006 p^{32} T^{34} + p^{36} T^{36} \) | |
| 53 | \( 1 - 28258 T^{2} + 376234951 T^{4} - 60099390728 p T^{6} + 19668594081682663 T^{8} - 96730600095142547238 T^{10} + \)\(40\!\cdots\!39\)\( T^{12} - \)\(14\!\cdots\!52\)\( T^{14} + \)\(48\!\cdots\!82\)\( T^{16} - \)\(14\!\cdots\!68\)\( T^{18} + \)\(48\!\cdots\!82\)\( p^{4} T^{20} - \)\(14\!\cdots\!52\)\( p^{8} T^{22} + \)\(40\!\cdots\!39\)\( p^{12} T^{24} - 96730600095142547238 p^{16} T^{26} + 19668594081682663 p^{20} T^{28} - 60099390728 p^{25} T^{30} + 376234951 p^{28} T^{32} - 28258 p^{32} T^{34} + p^{36} T^{36} \) | |
| 59 | \( 1 - 30836 T^{2} + 457094786 T^{4} - 4404606479210 T^{6} + 31486122740647951 T^{8} - \)\(18\!\cdots\!20\)\( T^{10} + \)\(87\!\cdots\!93\)\( T^{12} - \)\(37\!\cdots\!30\)\( T^{14} + \)\(14\!\cdots\!61\)\( T^{16} - \)\(51\!\cdots\!72\)\( T^{18} + \)\(14\!\cdots\!61\)\( p^{4} T^{20} - \)\(37\!\cdots\!30\)\( p^{8} T^{22} + \)\(87\!\cdots\!93\)\( p^{12} T^{24} - \)\(18\!\cdots\!20\)\( p^{16} T^{26} + 31486122740647951 p^{20} T^{28} - 4404606479210 p^{24} T^{30} + 457094786 p^{28} T^{32} - 30836 p^{32} T^{34} + p^{36} T^{36} \) | |
| 61 | \( ( 1 + 4 T + 22089 T^{2} + 120776 T^{3} + 229885360 T^{4} + 1047111488 T^{5} + 1531821118880 T^{6} + 4153434614200 T^{7} + 7423125474637742 T^{8} + 12831747182843000 T^{9} + 7423125474637742 p^{2} T^{10} + 4153434614200 p^{4} T^{11} + 1531821118880 p^{6} T^{12} + 1047111488 p^{8} T^{13} + 229885360 p^{10} T^{14} + 120776 p^{12} T^{15} + 22089 p^{14} T^{16} + 4 p^{16} T^{17} + p^{18} T^{18} )^{2} \) | |
| 67 | \( ( 1 - 68 T + 21814 T^{2} - 1189042 T^{3} + 244572607 T^{4} - 11786117484 T^{5} + 1856222695261 T^{6} - 80019959855790 T^{7} + 10625551371104613 T^{8} - 412616602813301792 T^{9} + 10625551371104613 p^{2} T^{10} - 80019959855790 p^{4} T^{11} + 1856222695261 p^{6} T^{12} - 11786117484 p^{8} T^{13} + 244572607 p^{10} T^{14} - 1189042 p^{12} T^{15} + 21814 p^{14} T^{16} - 68 p^{16} T^{17} + p^{18} T^{18} )^{2} \) | |
| 71 | \( 1 - 58426 T^{2} + 1599516201 T^{4} - 27059071191696 T^{6} + 4412897405334748 p T^{8} - \)\(25\!\cdots\!56\)\( T^{10} + \)\(15\!\cdots\!64\)\( T^{12} - \)\(63\!\cdots\!96\)\( T^{14} + \)\(18\!\cdots\!18\)\( T^{16} - \)\(55\!\cdots\!16\)\( T^{18} + \)\(18\!\cdots\!18\)\( p^{4} T^{20} - \)\(63\!\cdots\!96\)\( p^{8} T^{22} + \)\(15\!\cdots\!64\)\( p^{12} T^{24} - \)\(25\!\cdots\!56\)\( p^{16} T^{26} + 4412897405334748 p^{21} T^{28} - 27059071191696 p^{24} T^{30} + 1599516201 p^{28} T^{32} - 58426 p^{32} T^{34} + p^{36} T^{36} \) | |
| 73 | \( ( 1 + 34 T + 41561 T^{2} + 1306608 T^{3} + 796942948 T^{4} + 22657249656 T^{5} + 9241801068244 T^{6} + 231222504523408 T^{7} + 71360171680102526 T^{8} + 1518161092421699788 T^{9} + 71360171680102526 p^{2} T^{10} + 231222504523408 p^{4} T^{11} + 9241801068244 p^{6} T^{12} + 22657249656 p^{8} T^{13} + 796942948 p^{10} T^{14} + 1306608 p^{12} T^{15} + 41561 p^{14} T^{16} + 34 p^{16} T^{17} + p^{18} T^{18} )^{2} \) | |
| 79 | \( ( 1 + 48 T + 24665 T^{2} + 13354 p T^{3} + 300845409 T^{4} + 8686269614 T^{5} + 2453489836573 T^{6} + 38551769310754 T^{7} + 15849909493146088 T^{8} + 170275248322133524 T^{9} + 15849909493146088 p^{2} T^{10} + 38551769310754 p^{4} T^{11} + 2453489836573 p^{6} T^{12} + 8686269614 p^{8} T^{13} + 300845409 p^{10} T^{14} + 13354 p^{13} T^{15} + 24665 p^{14} T^{16} + 48 p^{16} T^{17} + p^{18} T^{18} )^{2} \) | |
| 83 | \( 1 - 49342 T^{2} + 1153709361 T^{4} - 17230738985856 T^{6} + 191377377084877148 T^{8} - \)\(17\!\cdots\!48\)\( T^{10} + \)\(15\!\cdots\!04\)\( T^{12} - \)\(12\!\cdots\!80\)\( T^{14} + \)\(97\!\cdots\!02\)\( T^{16} - \)\(69\!\cdots\!60\)\( T^{18} + \)\(97\!\cdots\!02\)\( p^{4} T^{20} - \)\(12\!\cdots\!80\)\( p^{8} T^{22} + \)\(15\!\cdots\!04\)\( p^{12} T^{24} - \)\(17\!\cdots\!48\)\( p^{16} T^{26} + 191377377084877148 p^{20} T^{28} - 17230738985856 p^{24} T^{30} + 1153709361 p^{28} T^{32} - 49342 p^{32} T^{34} + p^{36} T^{36} \) | |
| 89 | \( 1 - 90908 T^{2} + 4082745746 T^{4} - 120546782828082 T^{6} + 2627474481894923183 T^{8} - \)\(45\!\cdots\!12\)\( T^{10} + \)\(62\!\cdots\!33\)\( T^{12} - \)\(73\!\cdots\!30\)\( T^{14} + \)\(73\!\cdots\!53\)\( T^{16} - \)\(62\!\cdots\!28\)\( T^{18} + \)\(73\!\cdots\!53\)\( p^{4} T^{20} - \)\(73\!\cdots\!30\)\( p^{8} T^{22} + \)\(62\!\cdots\!33\)\( p^{12} T^{24} - \)\(45\!\cdots\!12\)\( p^{16} T^{26} + 2627474481894923183 p^{20} T^{28} - 120546782828082 p^{24} T^{30} + 4082745746 p^{28} T^{32} - 90908 p^{32} T^{34} + p^{36} T^{36} \) | |
| 97 | \( ( 1 + 6 T + 61557 T^{2} + 429008 T^{3} + 1795062012 T^{4} + 12770095408 T^{5} + 32892113119604 T^{6} + 221262888097072 T^{7} + 422410628729547994 T^{8} + 2521535824631667092 T^{9} + 422410628729547994 p^{2} T^{10} + 221262888097072 p^{4} T^{11} + 32892113119604 p^{6} T^{12} + 12770095408 p^{8} T^{13} + 1795062012 p^{10} T^{14} + 429008 p^{12} T^{15} + 61557 p^{14} T^{16} + 6 p^{16} T^{17} + p^{18} T^{18} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.77005744218312804810104770247, −3.62624834910911309340726103938, −3.44632078816175093103703748264, −3.35156851645513330035358047966, −3.28631845725135580558033893511, −3.22290086150977458253122487404, −3.20754855036388406287327894396, −3.19974464769321935663059570955, −2.92235869160431869154966533960, −2.78858850439680953257550876189, −2.74835802952246660502400426331, −2.73210941566446733115235700646, −2.63492293340654656418048907501, −2.23855473437582003629524144154, −2.23094328715608785118440747255, −2.15007229104252804862647919006, −2.14400527916633310263682194579, −1.80494444490225949097275272613, −1.69504813372663780385865762440, −1.54037337537877449204162841659, −1.23626821902947062873504699474, −1.15566539792745774505411324936, −1.09125916722000905698814640591, −0.71236737788214873756380602389, −0.04551199461203724061054901145, 0.04551199461203724061054901145, 0.71236737788214873756380602389, 1.09125916722000905698814640591, 1.15566539792745774505411324936, 1.23626821902947062873504699474, 1.54037337537877449204162841659, 1.69504813372663780385865762440, 1.80494444490225949097275272613, 2.14400527916633310263682194579, 2.15007229104252804862647919006, 2.23094328715608785118440747255, 2.23855473437582003629524144154, 2.63492293340654656418048907501, 2.73210941566446733115235700646, 2.74835802952246660502400426331, 2.78858850439680953257550876189, 2.92235869160431869154966533960, 3.19974464769321935663059570955, 3.20754855036388406287327894396, 3.22290086150977458253122487404, 3.28631845725135580558033893511, 3.35156851645513330035358047966, 3.44632078816175093103703748264, 3.62624834910911309340726103938, 3.77005744218312804810104770247
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.