Dirichlet series
| L(s) = 1 | + 8·7-s − 18·9-s + 16·11-s + 64·23-s − 30·25-s + 104·29-s + 32·37-s − 152·43-s + 62·49-s + 176·53-s − 144·63-s − 168·67-s − 32·71-s + 128·77-s − 120·79-s + 189·81-s − 288·99-s − 416·107-s − 376·109-s − 400·113-s − 204·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 8/7·7-s − 2·9-s + 1.45·11-s + 2.78·23-s − 6/5·25-s + 3.58·29-s + 0.864·37-s − 3.53·43-s + 1.26·49-s + 3.32·53-s − 2.28·63-s − 2.50·67-s − 0.450·71-s + 1.66·77-s − 1.51·79-s + 7/3·81-s − 2.90·99-s − 3.88·107-s − 3.44·109-s − 3.53·113-s − 1.68·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.46704\times 10^{19}\) |
| Root analytic conductor: | \(6.76584\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.4893538093\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4893538093\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T^{2} )^{6} \) | |
| 5 | \( ( 1 + p T^{2} )^{6} \) | |
| 7 | \( 1 - 8 T + 2 T^{2} - 312 T^{3} + 4255 T^{4} - 1984 p T^{5} + 892 p^{2} T^{6} - 1984 p^{3} T^{7} + 4255 p^{4} T^{8} - 312 p^{6} T^{9} + 2 p^{8} T^{10} - 8 p^{10} T^{11} + p^{12} T^{12} \) | |
| good | 11 | \( ( 1 - 8 T + 18 p T^{2} - 1416 T^{3} + 30895 T^{4} - 264944 T^{5} + 5610292 T^{6} - 264944 p^{2} T^{7} + 30895 p^{4} T^{8} - 1416 p^{6} T^{9} + 18 p^{9} T^{10} - 8 p^{10} T^{11} + p^{12} T^{12} )^{2} \) |
| 13 | \( 1 - 852 T^{2} + 436674 T^{4} - 159409348 T^{6} + 45042421839 T^{8} - 10253858128680 T^{10} + 1907667001388316 T^{12} - 10253858128680 p^{4} T^{14} + 45042421839 p^{8} T^{16} - 159409348 p^{12} T^{18} + 436674 p^{16} T^{20} - 852 p^{20} T^{22} + p^{24} T^{24} \) | |
| 17 | \( 1 - 2172 T^{2} + 141282 p T^{4} - 1757595148 T^{6} + 55392204927 p T^{8} - 387997181982840 T^{10} + 125896935467491356 T^{12} - 387997181982840 p^{4} T^{14} + 55392204927 p^{9} T^{16} - 1757595148 p^{12} T^{18} + 141282 p^{17} T^{20} - 2172 p^{20} T^{22} + p^{24} T^{24} \) | |
| 19 | \( 1 - 1404 T^{2} + 1342050 T^{4} - 896154316 T^{6} + 72919437 p^{3} T^{8} - 11983720660392 p T^{10} + 90028611036772572 T^{12} - 11983720660392 p^{5} T^{14} + 72919437 p^{11} T^{16} - 896154316 p^{12} T^{18} + 1342050 p^{16} T^{20} - 1404 p^{20} T^{22} + p^{24} T^{24} \) | |
| 23 | \( ( 1 - 32 T + 2418 T^{2} - 60144 T^{3} + 2694751 T^{4} - 53735792 T^{5} + 1782680284 T^{6} - 53735792 p^{2} T^{7} + 2694751 p^{4} T^{8} - 60144 p^{6} T^{9} + 2418 p^{8} T^{10} - 32 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 29 | \( ( 1 - 52 T + 3990 T^{2} - 132324 T^{3} + 6311983 T^{4} - 164304136 T^{5} + 6334525012 T^{6} - 164304136 p^{2} T^{7} + 6311983 p^{4} T^{8} - 132324 p^{6} T^{9} + 3990 p^{8} T^{10} - 52 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 31 | \( 1 - 6228 T^{2} + 19622274 T^{4} - 42341564932 T^{6} + 69647410996719 T^{8} - 91080823295899560 T^{10} + 96735996796679061276 T^{12} - 91080823295899560 p^{4} T^{14} + 69647410996719 p^{8} T^{16} - 42341564932 p^{12} T^{18} + 19622274 p^{16} T^{20} - 6228 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( ( 1 - 16 T + 4622 T^{2} + 32784 T^{3} + 7136719 T^{4} + 278279680 T^{5} + 7429424740 T^{6} + 278279680 p^{2} T^{7} + 7136719 p^{4} T^{8} + 32784 p^{6} T^{9} + 4622 p^{8} T^{10} - 16 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 41 | \( 1 - 7524 T^{2} + 32464386 T^{4} - 93240551380 T^{6} + 206850565610415 T^{8} - 379921653079011144 T^{10} + \)\(65\!\cdots\!56\)\( T^{12} - 379921653079011144 p^{4} T^{14} + 206850565610415 p^{8} T^{16} - 93240551380 p^{12} T^{18} + 32464386 p^{16} T^{20} - 7524 p^{20} T^{22} + p^{24} T^{24} \) | |
| 43 | \( ( 1 + 76 T + 9590 T^{2} + 631788 T^{3} + 42146383 T^{4} + 2188827368 T^{5} + 102971644948 T^{6} + 2188827368 p^{2} T^{7} + 42146383 p^{4} T^{8} + 631788 p^{6} T^{9} + 9590 p^{8} T^{10} + 76 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 47 | \( 1 - 6132 T^{2} + 27456354 T^{4} - 92403901348 T^{6} + 277727516070639 T^{8} - 754801362269230440 T^{10} + \)\(17\!\cdots\!96\)\( T^{12} - 754801362269230440 p^{4} T^{14} + 277727516070639 p^{8} T^{16} - 92403901348 p^{12} T^{18} + 27456354 p^{16} T^{20} - 6132 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( ( 1 - 88 T + 10434 T^{2} - 574440 T^{3} + 50272015 T^{4} - 2575931008 T^{5} + 181692199804 T^{6} - 2575931008 p^{2} T^{7} + 50272015 p^{4} T^{8} - 574440 p^{6} T^{9} + 10434 p^{8} T^{10} - 88 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 59 | \( 1 - 15948 T^{2} + 137050242 T^{4} - 864152710108 T^{6} + 4419517662043791 T^{8} - 18954945757384385304 T^{10} + \)\(70\!\cdots\!16\)\( T^{12} - 18954945757384385304 p^{4} T^{14} + 4419517662043791 p^{8} T^{16} - 864152710108 p^{12} T^{18} + 137050242 p^{16} T^{20} - 15948 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( 1 - 13716 T^{2} + 109224834 T^{4} - 648593370628 T^{6} + 3203041286245839 T^{8} - 13994290873734287016 T^{10} + \)\(55\!\cdots\!56\)\( T^{12} - 13994290873734287016 p^{4} T^{14} + 3203041286245839 p^{8} T^{16} - 648593370628 p^{12} T^{18} + 109224834 p^{16} T^{20} - 13716 p^{20} T^{22} + p^{24} T^{24} \) | |
| 67 | \( ( 1 + 84 T + 15366 T^{2} + 763540 T^{3} + 102124911 T^{4} + 4033313976 T^{5} + 514098700788 T^{6} + 4033313976 p^{2} T^{7} + 102124911 p^{4} T^{8} + 763540 p^{6} T^{9} + 15366 p^{8} T^{10} + 84 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 71 | \( ( 1 + 16 T + 16698 T^{2} + 42336 T^{3} + 131112895 T^{4} - 1041328304 T^{5} + 716701578892 T^{6} - 1041328304 p^{2} T^{7} + 131112895 p^{4} T^{8} + 42336 p^{6} T^{9} + 16698 p^{8} T^{10} + 16 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 73 | \( 1 - 19284 T^{2} + 2959410 p T^{4} - 1973102936452 T^{6} + 14880325491672495 T^{8} - 96746229105564519336 T^{10} + \)\(55\!\cdots\!08\)\( T^{12} - 96746229105564519336 p^{4} T^{14} + 14880325491672495 p^{8} T^{16} - 1973102936452 p^{12} T^{18} + 2959410 p^{17} T^{20} - 19284 p^{20} T^{22} + p^{24} T^{24} \) | |
| 79 | \( ( 1 + 60 T + 22242 T^{2} + 662476 T^{3} + 201003183 T^{4} + 1192106616 T^{5} + 1261845815388 T^{6} + 1192106616 p^{2} T^{7} + 201003183 p^{4} T^{8} + 662476 p^{6} T^{9} + 22242 p^{8} T^{10} + 60 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 83 | \( 1 - 45420 T^{2} + 1066844514 T^{4} - 17098817034364 T^{6} + 206563644037003983 T^{8} - \)\(19\!\cdots\!08\)\( T^{10} + \)\(15\!\cdots\!04\)\( T^{12} - \)\(19\!\cdots\!08\)\( p^{4} T^{14} + 206563644037003983 p^{8} T^{16} - 17098817034364 p^{12} T^{18} + 1066844514 p^{16} T^{20} - 45420 p^{20} T^{22} + p^{24} T^{24} \) | |
| 89 | \( 1 - 46020 T^{2} + 1126013634 T^{4} - 19221077499316 T^{6} + 252845241317038383 T^{8} - \)\(26\!\cdots\!72\)\( T^{10} + \)\(23\!\cdots\!84\)\( T^{12} - \)\(26\!\cdots\!72\)\( p^{4} T^{14} + 252845241317038383 p^{8} T^{16} - 19221077499316 p^{12} T^{18} + 1126013634 p^{16} T^{20} - 46020 p^{20} T^{22} + p^{24} T^{24} \) | |
| 97 | \( 1 - 51924 T^{2} + 1396974978 T^{4} - 26352399780484 T^{6} + 389015604150559215 T^{8} - \)\(47\!\cdots\!84\)\( T^{10} + \)\(48\!\cdots\!52\)\( T^{12} - \)\(47\!\cdots\!84\)\( p^{4} T^{14} + 389015604150559215 p^{8} T^{16} - 26352399780484 p^{12} T^{18} + 1396974978 p^{16} T^{20} - 51924 p^{20} T^{22} + p^{24} T^{24} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.66038322466872895884676131703, −2.53805958782001935078437754973, −2.52959738493888352595854171491, −2.51919289934668495543740566020, −2.50632634557836163893536624868, −2.47092172834042420222970187803, −2.05285083507307950809495239590, −1.97556354154976770148865594661, −1.93469485669518374672639242322, −1.76938824931931962026104543804, −1.61592428847380985862498426141, −1.60398512784772725357305637459, −1.57772445459358215603126418143, −1.42685219860972459842189600687, −1.24003720617806453485219885436, −1.20585741374104561937971404410, −0.971404846410572419043636044939, −0.963733923251450777028613531445, −0.918444489876943257203856623834, −0.883884685912972974108970281519, −0.846337423711938801495812971711, −0.40205013671143957531500145684, −0.17345569999265498074969917233, −0.13104080208839764914928281090, −0.082407087811333369522027848974, 0.082407087811333369522027848974, 0.13104080208839764914928281090, 0.17345569999265498074969917233, 0.40205013671143957531500145684, 0.846337423711938801495812971711, 0.883884685912972974108970281519, 0.918444489876943257203856623834, 0.963733923251450777028613531445, 0.971404846410572419043636044939, 1.20585741374104561937971404410, 1.24003720617806453485219885436, 1.42685219860972459842189600687, 1.57772445459358215603126418143, 1.60398512784772725357305637459, 1.61592428847380985862498426141, 1.76938824931931962026104543804, 1.93469485669518374672639242322, 1.97556354154976770148865594661, 2.05285083507307950809495239590, 2.47092172834042420222970187803, 2.50632634557836163893536624868, 2.51919289934668495543740566020, 2.52959738493888352595854171491, 2.53805958782001935078437754973, 2.66038322466872895884676131703
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.