Dirichlet series
| L(s) = 1 | + 10·3-s + 50·9-s + 36·11-s − 120·23-s + 600·25-s + 78·27-s + 360·33-s + 528·37-s − 1.24e3·47-s + 1.58e3·49-s + 2.50e3·59-s + 624·61-s − 1.20e3·69-s − 2.04e3·71-s − 216·73-s + 6.00e3·75-s − 739·81-s + 4.57e3·83-s − 48·97-s + 1.80e3·99-s + 6.49e3·107-s + 144·109-s + 5.28e3·111-s − 6.69e3·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.92·3-s + 1.85·9-s + 0.986·11-s − 1.08·23-s + 24/5·25-s + 0.555·27-s + 1.89·33-s + 2.34·37-s − 3.87·47-s + 4.61·49-s + 5.53·59-s + 1.30·61-s − 2.09·69-s − 3.40·71-s − 0.346·73-s + 9.23·75-s − 1.01·81-s + 6.04·83-s − 0.0502·97-s + 1.82·99-s + 5.86·107-s + 0.126·109-s + 4.51·111-s − 5.03·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{84} \cdot 3^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.82964\times 10^{16}\) |
| Root analytic conductor: | \(4.75990\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{84} \cdot 3^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(169.9208904\) |
| \(L(\frac12)\) | \(\approx\) | \(169.9208904\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 10 T + 50 T^{2} - 26 p T^{3} - 67 p T^{4} + 124 p^{3} T^{5} - 1924 p^{2} T^{6} + 124 p^{6} T^{7} - 67 p^{7} T^{8} - 26 p^{10} T^{9} + 50 p^{12} T^{10} - 10 p^{15} T^{11} + p^{18} T^{12} \) | |
| good | 5 | \( 1 - 24 p^{2} T^{2} + 36642 p T^{4} - 33731768 T^{6} + 3958679103 T^{8} - 292014000048 T^{10} + 22017716382796 T^{12} - 292014000048 p^{6} T^{14} + 3958679103 p^{12} T^{16} - 33731768 p^{18} T^{18} + 36642 p^{25} T^{20} - 24 p^{32} T^{22} + p^{36} T^{24} \) |
| 7 | \( 1 - 1584 T^{2} + 1166442 T^{4} - 478563952 T^{6} + 86370897327 T^{8} + 18087151507872 T^{10} - 14510145747466484 T^{12} + 18087151507872 p^{6} T^{14} + 86370897327 p^{12} T^{16} - 478563952 p^{18} T^{18} + 1166442 p^{24} T^{20} - 1584 p^{30} T^{22} + p^{36} T^{24} \) | |
| 11 | \( ( 1 - 18 T + 3834 T^{2} - 13302 T^{3} + 619797 p T^{4} + 39315780 T^{5} + 9651152012 T^{6} + 39315780 p^{3} T^{7} + 619797 p^{7} T^{8} - 13302 p^{9} T^{9} + 3834 p^{12} T^{10} - 18 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 13 | \( ( 1 + 5730 T^{2} + 3072 p T^{3} + 19329879 T^{4} + 32093184 T^{5} + 3911101356 p T^{6} + 32093184 p^{3} T^{7} + 19329879 p^{6} T^{8} + 3072 p^{10} T^{9} + 5730 p^{12} T^{10} + p^{18} T^{12} )^{2} \) | |
| 17 | \( 1 - 31164 T^{2} + 484190562 T^{4} - 4991368612172 T^{6} + 38572842772586415 T^{8} - \)\(24\!\cdots\!92\)\( T^{10} + \)\(12\!\cdots\!56\)\( T^{12} - \)\(24\!\cdots\!92\)\( p^{6} T^{14} + 38572842772586415 p^{12} T^{16} - 4991368612172 p^{18} T^{18} + 484190562 p^{24} T^{20} - 31164 p^{30} T^{22} + p^{36} T^{24} \) | |
| 19 | \( 1 - 38160 T^{2} + 808508346 T^{4} - 11916467660880 T^{6} + 135299522971061151 T^{8} - \)\(12\!\cdots\!08\)\( T^{10} + \)\(93\!\cdots\!44\)\( T^{12} - \)\(12\!\cdots\!08\)\( p^{6} T^{14} + 135299522971061151 p^{12} T^{16} - 11916467660880 p^{18} T^{18} + 808508346 p^{24} T^{20} - 38160 p^{30} T^{22} + p^{36} T^{24} \) | |
| 23 | \( ( 1 + 60 T + 35946 T^{2} + 1305844 T^{3} + 618033375 T^{4} + 10828626072 T^{5} + 7737521174604 T^{6} + 10828626072 p^{3} T^{7} + 618033375 p^{6} T^{8} + 1305844 p^{9} T^{9} + 35946 p^{12} T^{10} + 60 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 29 | \( 1 - 173784 T^{2} + 15604233546 T^{4} - 932142550689464 T^{6} + 41087275229333246751 T^{8} - \)\(14\!\cdots\!16\)\( T^{10} + \)\(38\!\cdots\!20\)\( T^{12} - \)\(14\!\cdots\!16\)\( p^{6} T^{14} + 41087275229333246751 p^{12} T^{16} - 932142550689464 p^{18} T^{18} + 15604233546 p^{24} T^{20} - 173784 p^{30} T^{22} + p^{36} T^{24} \) | |
| 31 | \( 1 - 188448 T^{2} + 548969046 p T^{4} - 999868768809888 T^{6} + 44197970839051250895 T^{8} - \)\(16\!\cdots\!56\)\( T^{10} + \)\(51\!\cdots\!24\)\( T^{12} - \)\(16\!\cdots\!56\)\( p^{6} T^{14} + 44197970839051250895 p^{12} T^{16} - 999868768809888 p^{18} T^{18} + 548969046 p^{25} T^{20} - 188448 p^{30} T^{22} + p^{36} T^{24} \) | |
| 37 | \( ( 1 - 264 T + 179778 T^{2} - 37541704 T^{3} + 13332243207 T^{4} - 2487709265808 T^{5} + 688884666730300 T^{6} - 2487709265808 p^{3} T^{7} + 13332243207 p^{6} T^{8} - 37541704 p^{9} T^{9} + 179778 p^{12} T^{10} - 264 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 41 | \( 1 - 472860 T^{2} + 113880883458 T^{4} - 18423203395045100 T^{6} + \)\(22\!\cdots\!39\)\( T^{8} - \)\(21\!\cdots\!48\)\( T^{10} + \)\(16\!\cdots\!96\)\( T^{12} - \)\(21\!\cdots\!48\)\( p^{6} T^{14} + \)\(22\!\cdots\!39\)\( p^{12} T^{16} - 18423203395045100 p^{18} T^{18} + 113880883458 p^{24} T^{20} - 472860 p^{30} T^{22} + p^{36} T^{24} \) | |
| 43 | \( 1 - 386256 T^{2} + 75858413850 T^{4} - 10754684841540112 T^{6} + \)\(12\!\cdots\!07\)\( T^{8} - \)\(12\!\cdots\!08\)\( T^{10} + \)\(10\!\cdots\!20\)\( T^{12} - \)\(12\!\cdots\!08\)\( p^{6} T^{14} + \)\(12\!\cdots\!07\)\( p^{12} T^{16} - 10754684841540112 p^{18} T^{18} + 75858413850 p^{24} T^{20} - 386256 p^{30} T^{22} + p^{36} T^{24} \) | |
| 47 | \( ( 1 + 624 T + 9222 p T^{2} + 110814928 T^{3} + 31731910959 T^{4} - 3786299059104 T^{5} - 661034171971540 T^{6} - 3786299059104 p^{3} T^{7} + 31731910959 p^{6} T^{8} + 110814928 p^{9} T^{9} + 9222 p^{13} T^{10} + 624 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 53 | \( 1 - 776088 T^{2} + 345553114986 T^{4} - 110076343120290296 T^{6} + \)\(27\!\cdots\!63\)\( T^{8} - \)\(54\!\cdots\!64\)\( T^{10} + \)\(88\!\cdots\!64\)\( T^{12} - \)\(54\!\cdots\!64\)\( p^{6} T^{14} + \)\(27\!\cdots\!63\)\( p^{12} T^{16} - 110076343120290296 p^{18} T^{18} + 345553114986 p^{24} T^{20} - 776088 p^{30} T^{22} + p^{36} T^{24} \) | |
| 59 | \( ( 1 - 1254 T + 1118754 T^{2} - 560306738 T^{3} + 186995225991 T^{4} - 14913042447924 T^{5} - 4616030684849220 T^{6} - 14913042447924 p^{3} T^{7} + 186995225991 p^{6} T^{8} - 560306738 p^{9} T^{9} + 1118754 p^{12} T^{10} - 1254 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 61 | \( ( 1 - 312 T + 8346 p T^{2} - 257561528 T^{3} + 211636873719 T^{4} - 86029951892976 T^{5} + 57402008417819932 T^{6} - 86029951892976 p^{3} T^{7} + 211636873719 p^{6} T^{8} - 257561528 p^{9} T^{9} + 8346 p^{13} T^{10} - 312 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 67 | \( 1 - 2286816 T^{2} + 2411853842778 T^{4} - 1566286389213019744 T^{6} + \)\(71\!\cdots\!43\)\( T^{8} - \)\(25\!\cdots\!84\)\( T^{10} + \)\(80\!\cdots\!64\)\( T^{12} - \)\(25\!\cdots\!84\)\( p^{6} T^{14} + \)\(71\!\cdots\!43\)\( p^{12} T^{16} - 1566286389213019744 p^{18} T^{18} + 2411853842778 p^{24} T^{20} - 2286816 p^{30} T^{22} + p^{36} T^{24} \) | |
| 71 | \( ( 1 + 1020 T + 2040810 T^{2} + 1572918772 T^{3} + 1765475696511 T^{4} + 1048763672220696 T^{5} + 832227063915098828 T^{6} + 1048763672220696 p^{3} T^{7} + 1765475696511 p^{6} T^{8} + 1572918772 p^{9} T^{9} + 2040810 p^{12} T^{10} + 1020 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 108 T + 839826 T^{2} + 159624796 T^{3} + 499086952191 T^{4} + 64393086787416 T^{5} + 241123492212399420 T^{6} + 64393086787416 p^{3} T^{7} + 499086952191 p^{6} T^{8} + 159624796 p^{9} T^{9} + 839826 p^{12} T^{10} + 108 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 79 | \( 1 - 4787328 T^{2} + 10929285260394 T^{4} - 15723000113198264960 T^{6} + \)\(15\!\cdots\!59\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(66\!\cdots\!56\)\( T^{12} - \)\(11\!\cdots\!88\)\( p^{6} T^{14} + \)\(15\!\cdots\!59\)\( p^{12} T^{16} - 15723000113198264960 p^{18} T^{18} + 10929285260394 p^{24} T^{20} - 4787328 p^{30} T^{22} + p^{36} T^{24} \) | |
| 83 | \( ( 1 - 2286 T + 4665594 T^{2} - 6371132122 T^{3} + 7586066763543 T^{4} - 7182264432963204 T^{5} + 5999536723098321228 T^{6} - 7182264432963204 p^{3} T^{7} + 7586066763543 p^{6} T^{8} - 6371132122 p^{9} T^{9} + 4665594 p^{12} T^{10} - 2286 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 89 | \( 1 - 4768524 T^{2} + 12015644253474 T^{4} - 230672060265750012 p T^{6} + \)\(26\!\cdots\!15\)\( T^{8} - \)\(25\!\cdots\!96\)\( T^{10} + \)\(20\!\cdots\!60\)\( T^{12} - \)\(25\!\cdots\!96\)\( p^{6} T^{14} + \)\(26\!\cdots\!15\)\( p^{12} T^{16} - 230672060265750012 p^{19} T^{18} + 12015644253474 p^{24} T^{20} - 4768524 p^{30} T^{22} + p^{36} T^{24} \) | |
| 97 | \( ( 1 + 24 T + 3053178 T^{2} + 1259266232 T^{3} + 4503411377103 T^{4} + 2753933866197552 T^{5} + 48632588768677228 p T^{6} + 2753933866197552 p^{3} T^{7} + 4503411377103 p^{6} T^{8} + 1259266232 p^{9} T^{9} + 3053178 p^{12} T^{10} + 24 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.30592391466596761453424867991, −3.11940765614824515467016516537, −3.05890986340864179416246456548, −3.00314582814298782128940102099, −2.79448903512730261150043278178, −2.68315697911434289786618930289, −2.58095345392586171903929221782, −2.50159993461140881264264914740, −2.48561398859408122833865195196, −2.23875338887436201034434318384, −2.19588458748667006014183438804, −2.09612469794709948245443154090, −1.94692426535217421073659798049, −1.94687676387648752931694246778, −1.55009372673397896983237810779, −1.52198085143425440145106268030, −1.16564884307451465270989802327, −1.14635855814090701065738052842, −1.12153956736246698794356243912, −1.08529974382968151725005444470, −0.77017547506056586787946000948, −0.56200649776354457573490389071, −0.47857036106700425450974519337, −0.46233952856605918639492961082, −0.30126268005233206738770217167, 0.30126268005233206738770217167, 0.46233952856605918639492961082, 0.47857036106700425450974519337, 0.56200649776354457573490389071, 0.77017547506056586787946000948, 1.08529974382968151725005444470, 1.12153956736246698794356243912, 1.14635855814090701065738052842, 1.16564884307451465270989802327, 1.52198085143425440145106268030, 1.55009372673397896983237810779, 1.94687676387648752931694246778, 1.94692426535217421073659798049, 2.09612469794709948245443154090, 2.19588458748667006014183438804, 2.23875338887436201034434318384, 2.48561398859408122833865195196, 2.50159993461140881264264914740, 2.58095345392586171903929221782, 2.68315697911434289786618930289, 2.79448903512730261150043278178, 3.00314582814298782128940102099, 3.05890986340864179416246456548, 3.11940765614824515467016516537, 3.30592391466596761453424867991
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.