Dirichlet series
| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s − 14·5-s + 8·6-s + 48·7-s − 44·8-s − 264·9-s + 28·10-s − 32·11-s − 8·12-s − 96·14-s + 56·15-s + 347·16-s + 528·18-s + 732·19-s − 28·20-s − 192·21-s + 64·22-s + 176·24-s + 98·25-s + 1.26e3·27-s + 96·28-s + 4.18e3·29-s − 112·30-s − 3.46e3·31-s + 118·32-s + ⋯ |
| L(s) = 1 | − 1/2·2-s − 4/9·3-s + 1/8·4-s − 0.559·5-s + 2/9·6-s + 0.979·7-s − 0.687·8-s − 3.25·9-s + 7/25·10-s − 0.264·11-s − 0.0555·12-s − 0.489·14-s + 0.248·15-s + 1.35·16-s + 1.62·18-s + 2.02·19-s − 0.0699·20-s − 0.435·21-s + 0.132·22-s + 0.305·24-s + 0.156·25-s + 1.73·27-s + 6/49·28-s + 4.97·29-s − 0.124·30-s − 3.60·31-s + 0.115·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4826809 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4826809 ^{s/2} \, \Gamma_{\C}(s+2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(4826809\) = \(13^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.88879\) |
| Root analytic conductor: | \(1.15922\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 4826809,\ (\ :[2]^{6}),\ 1)\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(0.6335870689\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6335870689\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 13 | \( 1 + 81 p^{2} T^{2} - 176 p^{4} T^{3} + 81 p^{6} T^{4} + p^{12} T^{6} \) |
| good | 2 | \( 1 + p T + p T^{2} + 11 p^{2} T^{3} - 175 T^{4} - 581 p T^{5} - 503 p T^{6} - 581 p^{5} T^{7} - 175 p^{8} T^{8} + 11 p^{14} T^{9} + p^{17} T^{10} + p^{21} T^{11} + p^{24} T^{12} \) |
| 3 | \( ( 1 + 2 T + 46 p T^{2} + 20 p^{2} T^{3} + 46 p^{5} T^{4} + 2 p^{8} T^{5} + p^{12} T^{6} )^{2} \) | |
| 5 | \( 1 + 14 T + 98 T^{2} + 8726 T^{3} + 769304 T^{4} + 5342354 T^{5} + 37472702 T^{6} + 5342354 p^{4} T^{7} + 769304 p^{8} T^{8} + 8726 p^{12} T^{9} + 98 p^{16} T^{10} + 14 p^{20} T^{11} + p^{24} T^{12} \) | |
| 7 | \( 1 - 48 T + 1152 T^{2} - 39716 T^{3} - 5764800 T^{4} + 372144048 T^{5} - 10433184376 T^{6} + 372144048 p^{4} T^{7} - 5764800 p^{8} T^{8} - 39716 p^{12} T^{9} + 1152 p^{16} T^{10} - 48 p^{20} T^{11} + p^{24} T^{12} \) | |
| 11 | \( 1 + 32 T + 512 T^{2} + 344936 T^{3} - 63068881 T^{4} - 504402488 p T^{5} - 708826336 p^{2} T^{6} - 504402488 p^{5} T^{7} - 63068881 p^{8} T^{8} + 344936 p^{12} T^{9} + 512 p^{16} T^{10} + 32 p^{20} T^{11} + p^{24} T^{12} \) | |
| 17 | \( 1 - 182400 T^{2} + 31740762624 T^{4} - 2752934184177374 T^{6} + 31740762624 p^{8} T^{8} - 182400 p^{16} T^{10} + p^{24} T^{12} \) | |
| 19 | \( 1 - 732 T + 267912 T^{2} - 58326044 T^{3} + 980477859 T^{4} + 1915050167496 T^{5} + 36465197577488 T^{6} + 1915050167496 p^{4} T^{7} + 980477859 p^{8} T^{8} - 58326044 p^{12} T^{9} + 267912 p^{16} T^{10} - 732 p^{20} T^{11} + p^{24} T^{12} \) | |
| 23 | \( 1 - 1524906 T^{2} + 1007971632387 T^{4} - 369112562746452020 T^{6} + 1007971632387 p^{8} T^{8} - 1524906 p^{16} T^{10} + p^{24} T^{12} \) | |
| 29 | \( ( 1 - 2092 T + 2568557 T^{2} - 2263548328 T^{3} + 2568557 p^{4} T^{4} - 2092 p^{8} T^{5} + p^{12} T^{6} )^{2} \) | |
| 31 | \( 1 + 3468 T + 6013512 T^{2} + 8766749740 T^{3} + 12186993284643 T^{4} + 14076253382920248 T^{5} + 13957766872742357648 T^{6} + 14076253382920248 p^{4} T^{7} + 12186993284643 p^{8} T^{8} + 8766749740 p^{12} T^{9} + 6013512 p^{16} T^{10} + 3468 p^{20} T^{11} + p^{24} T^{12} \) | |
| 37 | \( 1 + 1758 T + 1545282 T^{2} + 3527449702 T^{3} + 9479025269208 T^{4} + 9214471485611970 T^{5} + 7772724445723511006 T^{6} + 9214471485611970 p^{4} T^{7} + 9479025269208 p^{8} T^{8} + 3527449702 p^{12} T^{9} + 1545282 p^{16} T^{10} + 1758 p^{20} T^{11} + p^{24} T^{12} \) | |
| 41 | \( 1 - 4750 T + 11281250 T^{2} - 23149979278 T^{3} + 37142272482563 T^{4} - 45662503590367708 T^{5} + 65846400896247469892 T^{6} - 45662503590367708 p^{4} T^{7} + 37142272482563 p^{8} T^{8} - 23149979278 p^{12} T^{9} + 11281250 p^{16} T^{10} - 4750 p^{20} T^{11} + p^{24} T^{12} \) | |
| 43 | \( 1 - 5149872 T^{2} + 18545289223032 T^{4} - 84377003006781422498 T^{6} + 18545289223032 p^{8} T^{8} - 5149872 p^{16} T^{10} + p^{24} T^{12} \) | |
| 47 | \( 1 + 6872 T + 23612192 T^{2} + 59503259324 T^{3} + 95575957382048 T^{4} + 82680226895549288 T^{5} + 81739598027076446408 T^{6} + 82680226895549288 p^{4} T^{7} + 95575957382048 p^{8} T^{8} + 59503259324 p^{12} T^{9} + 23612192 p^{16} T^{10} + 6872 p^{20} T^{11} + p^{24} T^{12} \) | |
| 53 | \( ( 1 - 1054 T + 11036969 T^{2} - 24267726304 T^{3} + 11036969 p^{4} T^{4} - 1054 p^{8} T^{5} + p^{12} T^{6} )^{2} \) | |
| 59 | \( 1 - 4372 T + 9557192 T^{2} + 8760544940 T^{3} - 22022350167805 T^{4} - 699590778075744808 T^{5} + \)\(33\!\cdots\!36\)\( T^{6} - 699590778075744808 p^{4} T^{7} - 22022350167805 p^{8} T^{8} + 8760544940 p^{12} T^{9} + 9557192 p^{16} T^{10} - 4372 p^{20} T^{11} + p^{24} T^{12} \) | |
| 61 | \( ( 1 - 2994 T + 39378039 T^{2} - 81711006860 T^{3} + 39378039 p^{4} T^{4} - 2994 p^{8} T^{5} + p^{12} T^{6} )^{2} \) | |
| 67 | \( 1 - 72 T + 2592 T^{2} + 2528502352 T^{3} + 569584534669983 T^{4} - 216259992437699016 T^{5} + 17291018413684499168 T^{6} - 216259992437699016 p^{4} T^{7} + 569584534669983 p^{8} T^{8} + 2528502352 p^{12} T^{9} + 2592 p^{16} T^{10} - 72 p^{20} T^{11} + p^{24} T^{12} \) | |
| 71 | \( 1 + 14672 T + 107633792 T^{2} + 664874016500 T^{3} + 4498446529994480 T^{4} + 28160075410127913488 T^{5} + \)\(15\!\cdots\!76\)\( T^{6} + 28160075410127913488 p^{4} T^{7} + 4498446529994480 p^{8} T^{8} + 664874016500 p^{12} T^{9} + 107633792 p^{16} T^{10} + 14672 p^{20} T^{11} + p^{24} T^{12} \) | |
| 73 | \( 1 - 5874 T + 17251938 T^{2} - 193617003890 T^{3} + 3002766228450819 T^{4} - 10485445128171427236 T^{5} + \)\(28\!\cdots\!92\)\( T^{6} - 10485445128171427236 p^{4} T^{7} + 3002766228450819 p^{8} T^{8} - 193617003890 p^{12} T^{9} + 17251938 p^{16} T^{10} - 5874 p^{20} T^{11} + p^{24} T^{12} \) | |
| 79 | \( ( 1 - 1308 T + 107661369 T^{2} - 109363076192 T^{3} + 107661369 p^{4} T^{4} - 1308 p^{8} T^{5} + p^{12} T^{6} )^{2} \) | |
| 83 | \( 1 - 19264 T + 185550848 T^{2} - 1498697581912 T^{3} + 8779394341391855 T^{4} - 36613623736591560520 T^{5} + \)\(19\!\cdots\!12\)\( T^{6} - 36613623736591560520 p^{4} T^{7} + 8779394341391855 p^{8} T^{8} - 1498697581912 p^{12} T^{9} + 185550848 p^{16} T^{10} - 19264 p^{20} T^{11} + p^{24} T^{12} \) | |
| 89 | \( 1 + 986 T + 486098 T^{2} - 154474746070 T^{3} + 1975473144822275 T^{4} + 28274168611850630084 T^{5} + \)\(38\!\cdots\!24\)\( T^{6} + 28274168611850630084 p^{4} T^{7} + 1975473144822275 p^{8} T^{8} - 154474746070 p^{12} T^{9} + 486098 p^{16} T^{10} + 986 p^{20} T^{11} + p^{24} T^{12} \) | |
| 97 | \( 1 + 23154 T + 268053858 T^{2} + 2366383285666 T^{3} + 18858271611968655 T^{4} + \)\(17\!\cdots\!56\)\( T^{5} + \)\(18\!\cdots\!12\)\( T^{6} + \)\(17\!\cdots\!56\)\( p^{4} T^{7} + 18858271611968655 p^{8} T^{8} + 2366383285666 p^{12} T^{9} + 268053858 p^{16} T^{10} + 23154 p^{20} T^{11} + p^{24} T^{12} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.18732214350939932892789591931, −11.18225639000555647234326714589, −10.84941116658297932101209745926, −10.70743721140277450898310539819, −10.06059520603840747602655506567, −9.867523882550041331979556102007, −9.697541005144794111392895199832, −8.965133156006368241761723611016, −8.912159148923288794444411086608, −8.581337885125100114937737405767, −8.255486743805247326931621823863, −8.213166794039858770941057051581, −7.84481413203184190526040178581, −7.30050391189314276652359369699, −7.06802492569208739811394267935, −6.30947448871213233121168780664, −6.21978837160010328861730921704, −5.52959134200637178627361099110, −5.44033687431510483445844758930, −5.15852910004743534924312568457, −4.55405711746080778770261517914, −3.47382028914798590928897245272, −3.10968951041503230963672299874, −2.65095457707160914672818633496, −0.71423285713753686510435011964, 0.71423285713753686510435011964, 2.65095457707160914672818633496, 3.10968951041503230963672299874, 3.47382028914798590928897245272, 4.55405711746080778770261517914, 5.15852910004743534924312568457, 5.44033687431510483445844758930, 5.52959134200637178627361099110, 6.21978837160010328861730921704, 6.30947448871213233121168780664, 7.06802492569208739811394267935, 7.30050391189314276652359369699, 7.84481413203184190526040178581, 8.213166794039858770941057051581, 8.255486743805247326931621823863, 8.581337885125100114937737405767, 8.912159148923288794444411086608, 8.965133156006368241761723611016, 9.697541005144794111392895199832, 9.867523882550041331979556102007, 10.06059520603840747602655506567, 10.70743721140277450898310539819, 10.84941116658297932101209745926, 11.18225639000555647234326714589, 11.18732214350939932892789591931
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.