Dirichlet series
| L(s) = 1 | − 48·2-s + 162·3-s + 1.34e3·4-s − 43·5-s − 7.77e3·6-s + 2.05e3·7-s − 2.86e4·8-s + 1.53e4·9-s + 2.06e3·10-s + 7.37e3·11-s + 2.17e5·12-s − 1.31e4·13-s − 9.87e4·14-s − 6.96e3·15-s + 5.16e5·16-s + 7.95e3·17-s − 7.34e5·18-s − 5.71e4·19-s − 5.77e4·20-s + 3.33e5·21-s − 3.53e5·22-s + 3.17e4·23-s − 4.64e6·24-s − 1.65e5·25-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 − 0.153·5-s − 14.6·6-s + 2.26·7-s − 19.7·8-s + 7·9-s + 0.652·10-s + 1.66·11-s + 36.3·12-s − 1.66·13-s − 9.62·14-s − 0.532·15-s + 63/2·16-s + 0.392·17-s − 29.6·18-s − 1.91·19-s − 1.61·20-s + 7.85·21-s − 7.08·22-s + 0.544·23-s − 68.5·24-s − 2.11·25-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
| \(L(4)\) | \(\approx\) | \(47.52581923\) |
| \(L(\frac12)\) | \(\approx\) | \(47.52581923\) |
| \(L(\frac{9}{2})\) | 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 | \( 1 + 43 T + 167439 T^{2} + 10974104 T^{3} + 3295288109 p T^{4} + 64245392161 p^{2} T^{5} + 11844274566814 p^{3} T^{6} + 64245392161 p^{9} T^{7} + 3295288109 p^{15} T^{8} + 10974104 p^{21} T^{9} + 167439 p^{28} T^{10} + 43 p^{35} T^{11} + p^{42} T^{12} \) |
| 11 | \( 1 - 670 p T + 6442983 p T^{2} - 247779255076 T^{3} + 1380145711201567 T^{4} - 1333369743019177274 T^{5} + \)\(14\!\cdots\!18\)\( T^{6} - 1333369743019177274 p^{7} T^{7} + 1380145711201567 p^{14} T^{8} - 247779255076 p^{21} T^{9} + 6442983 p^{29} T^{10} - 670 p^{36} T^{11} + p^{42} T^{12} \) | |
| 17 | \( 1 - 7950 T + 452977633 T^{2} - 4234496544240 T^{3} + 310752983446969959 T^{4} - \)\(40\!\cdots\!42\)\( T^{5} + \)\(12\!\cdots\!26\)\( T^{6} - \)\(40\!\cdots\!42\)\( p^{7} T^{7} + 310752983446969959 p^{14} T^{8} - 4234496544240 p^{21} T^{9} + 452977633 p^{28} T^{10} - 7950 p^{35} T^{11} + p^{42} T^{12} \) | |
| 19 | \( 1 + 57145 T + 3366753633 T^{2} + 129965000571666 T^{3} + 4341074555510634415 T^{4} + \)\(12\!\cdots\!25\)\( T^{5} + \)\(38\!\cdots\!70\)\( T^{6} + \)\(12\!\cdots\!25\)\( p^{7} T^{7} + 4341074555510634415 p^{14} T^{8} + 129965000571666 p^{21} T^{9} + 3366753633 p^{28} T^{10} + 57145 p^{35} T^{11} + p^{42} T^{12} \) | |
| 23 | \( 1 - 31769 T + 11056266301 T^{2} - 522069485463050 T^{3} + 70182243917654148287 T^{4} - \)\(28\!\cdots\!69\)\( T^{5} + \)\(30\!\cdots\!42\)\( T^{6} - \)\(28\!\cdots\!69\)\( p^{7} T^{7} + 70182243917654148287 p^{14} T^{8} - 522069485463050 p^{21} T^{9} + 11056266301 p^{28} T^{10} - 31769 p^{35} T^{11} + p^{42} T^{12} \) | |
| 29 | \( 1 + 36455 T + 65673731827 T^{2} + 3277518694240436 T^{3} + \)\(22\!\cdots\!49\)\( T^{4} + \)\(10\!\cdots\!01\)\( T^{5} + \)\(47\!\cdots\!34\)\( T^{6} + \)\(10\!\cdots\!01\)\( p^{7} T^{7} + \)\(22\!\cdots\!49\)\( p^{14} T^{8} + 3277518694240436 p^{21} T^{9} + 65673731827 p^{28} T^{10} + 36455 p^{35} T^{11} + p^{42} T^{12} \) | |
| 31 | \( 1 - 215069 T + 89238564686 T^{2} - 13746772329923007 T^{3} + \)\(37\!\cdots\!75\)\( T^{4} - \)\(51\!\cdots\!14\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} - \)\(51\!\cdots\!14\)\( p^{7} T^{7} + \)\(37\!\cdots\!75\)\( p^{14} T^{8} - 13746772329923007 p^{21} T^{9} + 89238564686 p^{28} T^{10} - 215069 p^{35} T^{11} + p^{42} T^{12} \) | |
| 37 | \( 1 - 133074 T + 308784607389 T^{2} - 52090523454521804 T^{3} + \)\(48\!\cdots\!59\)\( T^{4} - \)\(92\!\cdots\!70\)\( T^{5} + \)\(54\!\cdots\!50\)\( T^{6} - \)\(92\!\cdots\!70\)\( p^{7} T^{7} + \)\(48\!\cdots\!59\)\( p^{14} T^{8} - 52090523454521804 p^{21} T^{9} + 308784607389 p^{28} T^{10} - 133074 p^{35} T^{11} + p^{42} T^{12} \) | |
| 41 | \( 1 - 516452 T + 838244841202 T^{2} - 363993286186981556 T^{3} + \)\(32\!\cdots\!91\)\( T^{4} - \)\(11\!\cdots\!96\)\( T^{5} + \)\(78\!\cdots\!52\)\( T^{6} - \)\(11\!\cdots\!96\)\( p^{7} T^{7} + \)\(32\!\cdots\!91\)\( p^{14} T^{8} - 363993286186981556 p^{21} T^{9} + 838244841202 p^{28} T^{10} - 516452 p^{35} T^{11} + p^{42} T^{12} \) | |
| 43 | \( 1 + 3085 T + 846069227173 T^{2} - 78548306280925066 T^{3} + \)\(36\!\cdots\!27\)\( T^{4} - \)\(62\!\cdots\!39\)\( T^{5} + \)\(11\!\cdots\!18\)\( T^{6} - \)\(62\!\cdots\!39\)\( p^{7} T^{7} + \)\(36\!\cdots\!27\)\( p^{14} T^{8} - 78548306280925066 p^{21} T^{9} + 846069227173 p^{28} T^{10} + 3085 p^{35} T^{11} + p^{42} T^{12} \) | |
| 47 | \( 1 - 1463947 T + 2646873328206 T^{2} - 2434420193591853125 T^{3} + \)\(25\!\cdots\!83\)\( T^{4} - \)\(17\!\cdots\!54\)\( T^{5} + \)\(15\!\cdots\!92\)\( T^{6} - \)\(17\!\cdots\!54\)\( p^{7} T^{7} + \)\(25\!\cdots\!83\)\( p^{14} T^{8} - 2434420193591853125 p^{21} T^{9} + 2646873328206 p^{28} T^{10} - 1463947 p^{35} T^{11} + p^{42} T^{12} \) | |
| 53 | \( 1 + 1344571 T + 4988476392080 T^{2} + 6840772845853746803 T^{3} + \)\(12\!\cdots\!67\)\( T^{4} + \)\(14\!\cdots\!46\)\( T^{5} + \)\(17\!\cdots\!40\)\( T^{6} + \)\(14\!\cdots\!46\)\( p^{7} T^{7} + \)\(12\!\cdots\!67\)\( p^{14} T^{8} + 6840772845853746803 p^{21} T^{9} + 4988476392080 p^{28} T^{10} + 1344571 p^{35} T^{11} + p^{42} T^{12} \) | |
| 59 | \( 1 - 1810408 T + 9214421040822 T^{2} - 12746006806883037800 T^{3} + \)\(40\!\cdots\!67\)\( T^{4} - \)\(43\!\cdots\!64\)\( T^{5} + \)\(11\!\cdots\!04\)\( T^{6} - \)\(43\!\cdots\!64\)\( p^{7} T^{7} + \)\(40\!\cdots\!67\)\( p^{14} T^{8} - 12746006806883037800 p^{21} T^{9} + 9214421040822 p^{28} T^{10} - 1810408 p^{35} T^{11} + p^{42} T^{12} \) | |
| 61 | \( 1 - 4047390 T + 10654065129607 T^{2} - 20695293819589083978 T^{3} + \)\(34\!\cdots\!87\)\( T^{4} - \)\(53\!\cdots\!72\)\( T^{5} + \)\(95\!\cdots\!50\)\( T^{6} - \)\(53\!\cdots\!72\)\( p^{7} T^{7} + \)\(34\!\cdots\!87\)\( p^{14} T^{8} - 20695293819589083978 p^{21} T^{9} + 10654065129607 p^{28} T^{10} - 4047390 p^{35} T^{11} + p^{42} T^{12} \) | |
| 67 | \( 1 - 2393614 T + 9933562030526 T^{2} - 316812264165536430 p T^{3} + \)\(88\!\cdots\!23\)\( T^{4} - \)\(12\!\cdots\!48\)\( T^{5} + \)\(46\!\cdots\!76\)\( T^{6} - \)\(12\!\cdots\!48\)\( p^{7} T^{7} + \)\(88\!\cdots\!23\)\( p^{14} T^{8} - 316812264165536430 p^{22} T^{9} + 9933562030526 p^{28} T^{10} - 2393614 p^{35} T^{11} + p^{42} T^{12} \) | |
| 71 | \( 1 - 10341084 T + 71777925859902 T^{2} - \)\(35\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(55\!\cdots\!92\)\( T^{5} + \)\(18\!\cdots\!40\)\( T^{6} - \)\(55\!\cdots\!92\)\( p^{7} T^{7} + \)\(15\!\cdots\!15\)\( p^{14} T^{8} - \)\(35\!\cdots\!36\)\( p^{21} T^{9} + 71777925859902 p^{28} T^{10} - 10341084 p^{35} T^{11} + p^{42} T^{12} \) | |
| 73 | \( 1 - 5180001 T + 31482731933191 T^{2} - 70143749572935497208 T^{3} + \)\(23\!\cdots\!09\)\( T^{4} + \)\(25\!\cdots\!49\)\( T^{5} - \)\(99\!\cdots\!34\)\( T^{6} + \)\(25\!\cdots\!49\)\( p^{7} T^{7} + \)\(23\!\cdots\!09\)\( p^{14} T^{8} - 70143749572935497208 p^{21} T^{9} + 31482731933191 p^{28} T^{10} - 5180001 p^{35} T^{11} + p^{42} T^{12} \) | |
| 79 | \( 1 - 4624979 T + 29328772666898 T^{2} - 73453713201488419725 T^{3} + \)\(27\!\cdots\!95\)\( T^{4} - \)\(99\!\cdots\!62\)\( T^{5} + \)\(42\!\cdots\!48\)\( T^{6} - \)\(99\!\cdots\!62\)\( p^{7} T^{7} + \)\(27\!\cdots\!95\)\( p^{14} T^{8} - 73453713201488419725 p^{21} T^{9} + 29328772666898 p^{28} T^{10} - 4624979 p^{35} T^{11} + p^{42} T^{12} \) | |
| 83 | \( 1 - 11892699 T + 125531381687338 T^{2} - \)\(62\!\cdots\!69\)\( T^{3} + \)\(30\!\cdots\!25\)\( p T^{4} + \)\(75\!\cdots\!02\)\( T^{5} - \)\(11\!\cdots\!68\)\( T^{6} + \)\(75\!\cdots\!02\)\( p^{7} T^{7} + \)\(30\!\cdots\!25\)\( p^{15} T^{8} - \)\(62\!\cdots\!69\)\( p^{21} T^{9} + 125531381687338 p^{28} T^{10} - 11892699 p^{35} T^{11} + p^{42} T^{12} \) | |
| 89 | \( 1 - 9781713 T + 151552706384404 T^{2} - \)\(13\!\cdots\!21\)\( T^{3} + \)\(13\!\cdots\!55\)\( p T^{4} - \)\(89\!\cdots\!06\)\( T^{5} + \)\(63\!\cdots\!52\)\( T^{6} - \)\(89\!\cdots\!06\)\( p^{7} T^{7} + \)\(13\!\cdots\!55\)\( p^{15} T^{8} - \)\(13\!\cdots\!21\)\( p^{21} T^{9} + 151552706384404 p^{28} T^{10} - 9781713 p^{35} T^{11} + p^{42} T^{12} \) | |
| 97 | \( 1 - 5202537 T + 250398284557276 T^{2} - \)\(19\!\cdots\!93\)\( T^{3} + \)\(33\!\cdots\!79\)\( T^{4} - \)\(29\!\cdots\!38\)\( T^{5} + \)\(31\!\cdots\!72\)\( T^{6} - \)\(29\!\cdots\!38\)\( p^{7} T^{7} + \)\(33\!\cdots\!79\)\( p^{14} T^{8} - \)\(19\!\cdots\!93\)\( p^{21} T^{9} + 250398284557276 p^{28} T^{10} - 5202537 p^{35} T^{11} + p^{42} T^{12} \) | |
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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
−4.62468293954688111452919835125, −4.01744408505230889760830806018, −3.93672687475199227763433238935, −3.92579251772812122737352586741, −3.83368520651523635282542928311, −3.79533309585980862175661521209, −3.76959889395914711682302460423, −2.94243314687110265348178127161, −2.93517689400129079748890594403, −2.86997251898482629739027851676, −2.49788536105083821595782133515, −2.48110024372493266460048391098, −2.44134082996618490209067342643, −2.05298672061276719578382917257, −1.92837205587409255836531146490, −1.88783668324006555734472652742, −1.66402966457206636809137788579, −1.66075344592837576216082039097, −1.56089353411444017873469008470, −0.878120644194415075509068911265, −0.849260917156261232667499882814, −0.816065115529438827972349334857, −0.59265922906405197684288496811, −0.50018597728293339324545780929, −0.43967678735356495568798026429, 0.43967678735356495568798026429, 0.50018597728293339324545780929, 0.59265922906405197684288496811, 0.816065115529438827972349334857, 0.849260917156261232667499882814, 0.878120644194415075509068911265, 1.56089353411444017873469008470, 1.66075344592837576216082039097, 1.66402966457206636809137788579, 1.88783668324006555734472652742, 1.92837205587409255836531146490, 2.05298672061276719578382917257, 2.44134082996618490209067342643, 2.48110024372493266460048391098, 2.49788536105083821595782133515, 2.86997251898482629739027851676, 2.93517689400129079748890594403, 2.94243314687110265348178127161, 3.76959889395914711682302460423, 3.79533309585980862175661521209, 3.83368520651523635282542928311, 3.92579251772812122737352586741, 3.93672687475199227763433238935, 4.01744408505230889760830806018, 4.62468293954688111452919835125
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.