Dirichlet series
| L(s) = 1 | + 48·2-s − 162·3-s + 1.34e3·4-s + 13·5-s − 7.77e3·6-s + 2.05e3·7-s + 2.86e4·8-s + 1.53e4·9-s + 624·10-s + 1.00e4·11-s − 2.17e5·12-s − 1.31e4·13-s + 9.87e4·14-s − 2.10e3·15-s + 5.16e5·16-s + 2.12e4·17-s + 7.34e5·18-s + 9.52e3·19-s + 1.74e4·20-s − 3.33e5·21-s + 4.82e5·22-s + 3.32e4·23-s − 4.64e6·24-s − 1.01e5·25-s − 6.32e5·26-s − 1.10e6·27-s + 2.76e6·28-s + ⋯ |
| L(s) = 1 | + 4.24·2-s − 3.46·3-s + 21/2·4-s + 0.0465·5-s − 14.6·6-s + 2.26·7-s + 19.7·8-s + 7·9-s + 0.197·10-s + 2.27·11-s − 36.3·12-s − 1.66·13-s + 9.62·14-s − 0.161·15-s + 63/2·16-s + 1.04·17-s + 29.6·18-s + 0.318·19-s + 0.488·20-s − 7.85·21-s + 9.66·22-s + 0.569·23-s − 68.5·24-s − 1.30·25-s − 7.06·26-s − 10.7·27-s + 23.8·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.46205\times 10^{13}\) |
| Root analytic conductor: | \(13.0599\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(577.9524640\) |
| \(L(\frac12)\) | \(\approx\) | \(577.9524640\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{3} T )^{6} \) |
| 3 | \( ( 1 + p^{3} T )^{6} \) | |
| 7 | \( ( 1 - p^{3} T )^{6} \) | |
| 13 | \( ( 1 + p^{3} T )^{6} \) | |
| good | 5 | \( 1 - 13 T + 101799 T^{2} - 793772 p^{2} T^{3} + 94345169 p^{3} T^{4} - 23766019079 p^{3} T^{5} + 987378001502 p^{4} T^{6} - 23766019079 p^{10} T^{7} + 94345169 p^{17} T^{8} - 793772 p^{23} T^{9} + 101799 p^{28} T^{10} - 13 p^{35} T^{11} + p^{42} T^{12} \) |
| 11 | \( 1 - 914 p T + 72627429 T^{2} - 306435538340 T^{3} + 1326237671985943 T^{4} - 5606073349159751374 T^{5} + \)\(27\!\cdots\!18\)\( T^{6} - 5606073349159751374 p^{7} T^{7} + 1326237671985943 p^{14} T^{8} - 306435538340 p^{21} T^{9} + 72627429 p^{28} T^{10} - 914 p^{36} T^{11} + p^{42} T^{12} \) | |
| 17 | \( 1 - 21222 T + 982974121 T^{2} - 14662754021904 T^{3} + 364149425480783175 T^{4} - \)\(29\!\cdots\!50\)\( T^{5} + \)\(89\!\cdots\!38\)\( T^{6} - \)\(29\!\cdots\!50\)\( p^{7} T^{7} + 364149425480783175 p^{14} T^{8} - 14662754021904 p^{21} T^{9} + 982974121 p^{28} T^{10} - 21222 p^{35} T^{11} + p^{42} T^{12} \) | |
| 19 | \( 1 - 9527 T + 3487078929 T^{2} - 45605468375694 T^{3} + 5750270509238932063 T^{4} - \)\(86\!\cdots\!07\)\( T^{5} + \)\(61\!\cdots\!70\)\( T^{6} - \)\(86\!\cdots\!07\)\( p^{7} T^{7} + 5750270509238932063 p^{14} T^{8} - 45605468375694 p^{21} T^{9} + 3487078929 p^{28} T^{10} - 9527 p^{35} T^{11} + p^{42} T^{12} \) | |
| 23 | \( 1 - 33229 T + 9882524953 T^{2} - 318862642428598 T^{3} + 64670844266303164223 T^{4} - \)\(17\!\cdots\!13\)\( T^{5} + \)\(25\!\cdots\!02\)\( T^{6} - \)\(17\!\cdots\!13\)\( p^{7} T^{7} + 64670844266303164223 p^{14} T^{8} - 318862642428598 p^{21} T^{9} + 9882524953 p^{28} T^{10} - 33229 p^{35} T^{11} + p^{42} T^{12} \) | |
| 29 | \( 1 - 174185 T + 67920571267 T^{2} - 9433859472488924 T^{3} + \)\(21\!\cdots\!97\)\( T^{4} - \)\(31\!\cdots\!07\)\( p^{2} T^{5} + \)\(46\!\cdots\!82\)\( T^{6} - \)\(31\!\cdots\!07\)\( p^{9} T^{7} + \)\(21\!\cdots\!97\)\( p^{14} T^{8} - 9433859472488924 p^{21} T^{9} + 67920571267 p^{28} T^{10} - 174185 p^{35} T^{11} + p^{42} T^{12} \) | |
| 31 | \( 1 - 119045 T + 34993311086 T^{2} + 2368117744637865 T^{3} + \)\(65\!\cdots\!71\)\( T^{4} + \)\(12\!\cdots\!62\)\( T^{5} + \)\(17\!\cdots\!32\)\( T^{6} + \)\(12\!\cdots\!62\)\( p^{7} T^{7} + \)\(65\!\cdots\!71\)\( p^{14} T^{8} + 2368117744637865 p^{21} T^{9} + 34993311086 p^{28} T^{10} - 119045 p^{35} T^{11} + p^{42} T^{12} \) | |
| 37 | \( 1 - 56562 T + 330247237821 T^{2} + 21902852590151716 T^{3} + \)\(46\!\cdots\!75\)\( T^{4} + \)\(78\!\cdots\!90\)\( T^{5} + \)\(46\!\cdots\!58\)\( T^{6} + \)\(78\!\cdots\!90\)\( p^{7} T^{7} + \)\(46\!\cdots\!75\)\( p^{14} T^{8} + 21902852590151716 p^{21} T^{9} + 330247237821 p^{28} T^{10} - 56562 p^{35} T^{11} + p^{42} T^{12} \) | |
| 41 | \( 1 - 101632 T + 863442028090 T^{2} - 111621650369946256 T^{3} + \)\(34\!\cdots\!55\)\( T^{4} - \)\(44\!\cdots\!48\)\( T^{5} + \)\(85\!\cdots\!20\)\( T^{6} - \)\(44\!\cdots\!48\)\( p^{7} T^{7} + \)\(34\!\cdots\!55\)\( p^{14} T^{8} - 111621650369946256 p^{21} T^{9} + 863442028090 p^{28} T^{10} - 101632 p^{35} T^{11} + p^{42} T^{12} \) | |
| 43 | \( 1 - 441323 T + 1095994751389 T^{2} - 553243743661893154 T^{3} + \)\(59\!\cdots\!39\)\( T^{4} - \)\(28\!\cdots\!19\)\( T^{5} + \)\(19\!\cdots\!10\)\( T^{6} - \)\(28\!\cdots\!19\)\( p^{7} T^{7} + \)\(59\!\cdots\!39\)\( p^{14} T^{8} - 553243743661893154 p^{21} T^{9} + 1095994751389 p^{28} T^{10} - 441323 p^{35} T^{11} + p^{42} T^{12} \) | |
| 47 | \( 1 + 892849 T + 2377989971898 T^{2} + 1669051866466183895 T^{3} + \)\(25\!\cdots\!11\)\( T^{4} + \)\(14\!\cdots\!82\)\( T^{5} + \)\(16\!\cdots\!08\)\( T^{6} + \)\(14\!\cdots\!82\)\( p^{7} T^{7} + \)\(25\!\cdots\!11\)\( p^{14} T^{8} + 1669051866466183895 p^{21} T^{9} + 2377989971898 p^{28} T^{10} + 892849 p^{35} T^{11} + p^{42} T^{12} \) | |
| 53 | \( 1 - 2093965 T + 3522551874824 T^{2} - 2135556521622929525 T^{3} + \)\(13\!\cdots\!83\)\( T^{4} + \)\(12\!\cdots\!50\)\( T^{5} - \)\(61\!\cdots\!48\)\( T^{6} + \)\(12\!\cdots\!50\)\( p^{7} T^{7} + \)\(13\!\cdots\!83\)\( p^{14} T^{8} - 2135556521622929525 p^{21} T^{9} + 3522551874824 p^{28} T^{10} - 2093965 p^{35} T^{11} + p^{42} T^{12} \) | |
| 59 | \( 1 + 136204 T + 12886367931678 T^{2} + 827273707472333204 T^{3} + \)\(73\!\cdots\!03\)\( T^{4} + \)\(22\!\cdots\!80\)\( T^{5} + \)\(23\!\cdots\!36\)\( T^{6} + \)\(22\!\cdots\!80\)\( p^{7} T^{7} + \)\(73\!\cdots\!03\)\( p^{14} T^{8} + 827273707472333204 p^{21} T^{9} + 12886367931678 p^{28} T^{10} + 136204 p^{35} T^{11} + p^{42} T^{12} \) | |
| 61 | \( 1 + 3989946 T + 21419362712479 T^{2} + 56677573087088631150 T^{3} + \)\(17\!\cdots\!07\)\( T^{4} + \)\(33\!\cdots\!68\)\( T^{5} + \)\(73\!\cdots\!58\)\( T^{6} + \)\(33\!\cdots\!68\)\( p^{7} T^{7} + \)\(17\!\cdots\!07\)\( p^{14} T^{8} + 56677573087088631150 p^{21} T^{9} + 21419362712479 p^{28} T^{10} + 3989946 p^{35} T^{11} + p^{42} T^{12} \) | |
| 67 | \( 1 + 2218250 T + 25837270897982 T^{2} + 54003858141907254558 T^{3} + \)\(32\!\cdots\!23\)\( T^{4} + \)\(58\!\cdots\!20\)\( T^{5} + \)\(24\!\cdots\!48\)\( T^{6} + \)\(58\!\cdots\!20\)\( p^{7} T^{7} + \)\(32\!\cdots\!23\)\( p^{14} T^{8} + 54003858141907254558 p^{21} T^{9} + 25837270897982 p^{28} T^{10} + 2218250 p^{35} T^{11} + p^{42} T^{12} \) | |
| 71 | \( 1 - 2045928 T + 33356493334398 T^{2} - 56087972419485962568 T^{3} + \)\(60\!\cdots\!75\)\( T^{4} - \)\(84\!\cdots\!56\)\( T^{5} + \)\(67\!\cdots\!96\)\( T^{6} - \)\(84\!\cdots\!56\)\( p^{7} T^{7} + \)\(60\!\cdots\!75\)\( p^{14} T^{8} - 56087972419485962568 p^{21} T^{9} + 33356493334398 p^{28} T^{10} - 2045928 p^{35} T^{11} + p^{42} T^{12} \) | |
| 73 | \( 1 + 8557479 T + 72485522515327 T^{2} + \)\(39\!\cdots\!56\)\( T^{3} + \)\(19\!\cdots\!57\)\( T^{4} + \)\(79\!\cdots\!13\)\( T^{5} + \)\(28\!\cdots\!26\)\( T^{6} + \)\(79\!\cdots\!13\)\( p^{7} T^{7} + \)\(19\!\cdots\!57\)\( p^{14} T^{8} + \)\(39\!\cdots\!56\)\( p^{21} T^{9} + 72485522515327 p^{28} T^{10} + 8557479 p^{35} T^{11} + p^{42} T^{12} \) | |
| 79 | \( 1 + 8559709 T + 88266006012290 T^{2} + \)\(51\!\cdots\!51\)\( T^{3} + \)\(33\!\cdots\!87\)\( T^{4} + \)\(16\!\cdots\!82\)\( T^{5} + \)\(82\!\cdots\!44\)\( T^{6} + \)\(16\!\cdots\!82\)\( p^{7} T^{7} + \)\(33\!\cdots\!87\)\( p^{14} T^{8} + \)\(51\!\cdots\!51\)\( p^{21} T^{9} + 88266006012290 p^{28} T^{10} + 8559709 p^{35} T^{11} + p^{42} T^{12} \) | |
| 83 | \( 1 - 2496351 T + 134238343302166 T^{2} - \)\(30\!\cdots\!17\)\( T^{3} + \)\(81\!\cdots\!19\)\( T^{4} - \)\(15\!\cdots\!90\)\( T^{5} + \)\(28\!\cdots\!16\)\( T^{6} - \)\(15\!\cdots\!90\)\( p^{7} T^{7} + \)\(81\!\cdots\!19\)\( p^{14} T^{8} - \)\(30\!\cdots\!17\)\( p^{21} T^{9} + 134238343302166 p^{28} T^{10} - 2496351 p^{35} T^{11} + p^{42} T^{12} \) | |
| 89 | \( 1 + 2446683 T + 184129480003576 T^{2} + \)\(33\!\cdots\!63\)\( T^{3} + \)\(15\!\cdots\!99\)\( T^{4} + \)\(21\!\cdots\!54\)\( T^{5} + \)\(78\!\cdots\!84\)\( T^{6} + \)\(21\!\cdots\!54\)\( p^{7} T^{7} + \)\(15\!\cdots\!99\)\( p^{14} T^{8} + \)\(33\!\cdots\!63\)\( p^{21} T^{9} + 184129480003576 p^{28} T^{10} + 2446683 p^{35} T^{11} + p^{42} T^{12} \) | |
| 97 | \( 1 - 5786889 T + 200686910655940 T^{2} - \)\(19\!\cdots\!93\)\( T^{3} + \)\(25\!\cdots\!15\)\( T^{4} - \)\(26\!\cdots\!50\)\( T^{5} + \)\(22\!\cdots\!80\)\( T^{6} - \)\(26\!\cdots\!50\)\( p^{7} T^{7} + \)\(25\!\cdots\!15\)\( p^{14} T^{8} - \)\(19\!\cdots\!93\)\( p^{21} T^{9} + 200686910655940 p^{28} T^{10} - 5786889 p^{35} T^{11} + p^{42} T^{12} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.59296353356323683692907862801, −4.34350668063614578785309862471, −4.33405054274153888041222678099, −4.29543132174734789620514758141, −4.25194403046935776601615743072, −4.22653887804723747733963001340, −3.90958205808362665305146842822, −3.45493344644848667042509871548, −3.29022422428420204370916785040, −3.11912767045798532668076529501, −2.96542236637073825965401374802, −2.85676578301845696064918408922, −2.84270768088735897565629248908, −1.96464194231332228145152216484, −1.96249410641756181039870745100, −1.83880163905175569366439445854, −1.80941045126208978724596991778, −1.77517713518585767883058880609, −1.67110483367858199849726440604, −1.06356830952566761265573049586, −0.844560406000184324002009457992, −0.796339897879458885890696262861, −0.76601322036625800602058739366, −0.53275661327916592176825222330, −0.39104131176921821859353698190, 0.39104131176921821859353698190, 0.53275661327916592176825222330, 0.76601322036625800602058739366, 0.796339897879458885890696262861, 0.844560406000184324002009457992, 1.06356830952566761265573049586, 1.67110483367858199849726440604, 1.77517713518585767883058880609, 1.80941045126208978724596991778, 1.83880163905175569366439445854, 1.96249410641756181039870745100, 1.96464194231332228145152216484, 2.84270768088735897565629248908, 2.85676578301845696064918408922, 2.96542236637073825965401374802, 3.11912767045798532668076529501, 3.29022422428420204370916785040, 3.45493344644848667042509871548, 3.90958205808362665305146842822, 4.22653887804723747733963001340, 4.25194403046935776601615743072, 4.29543132174734789620514758141, 4.33405054274153888041222678099, 4.34350668063614578785309862471, 4.59296353356323683692907862801
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.