Dirichlet series
| L(s) = 1 | + 128·2-s − 972·3-s + 1.02e4·4-s − 1.12e4·5-s − 1.24e5·6-s + 9.33e3·7-s + 6.55e5·8-s + 5.90e5·9-s − 1.44e6·10-s − 7.59e4·11-s − 9.95e6·12-s + 9.11e4·13-s + 1.19e6·14-s + 1.09e7·15-s + 3.67e7·16-s − 4.84e6·17-s + 7.55e7·18-s − 9.90e6·19-s − 1.15e8·20-s − 9.06e6·21-s − 9.71e6·22-s + 6.92e7·23-s − 6.37e8·24-s + 3.78e6·25-s + 1.16e7·26-s − 2.86e8·27-s + 9.55e7·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 5·4-s − 1.61·5-s − 6.53·6-s + 0.209·7-s + 7.07·8-s + 10/3·9-s − 4.56·10-s − 0.142·11-s − 11.5·12-s + 0.0680·13-s + 0.593·14-s + 3.72·15-s + 35/4·16-s − 0.826·17-s + 9.42·18-s − 0.917·19-s − 8.06·20-s − 0.484·21-s − 0.401·22-s + 2.24·23-s − 16.3·24-s + 0.0775·25-s + 0.192·26-s − 3.84·27-s + 1.04·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 19^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.88627\times 10^{7}\) |
| Root analytic conductor: | \(9.35901\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 19^{4} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )\) |
Particular Values
| \(L(6)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{13}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{5} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + p^{5} T )^{4} \) | |
| 19 | $C_1$ | \( ( 1 + p^{5} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 11276 T + 24672577 p T^{2} + 9064034662 p^{3} T^{3} + 72395547788984 p^{3} T^{4} + 9064034662 p^{14} T^{5} + 24672577 p^{23} T^{6} + 11276 p^{33} T^{7} + p^{44} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 9330 T + 3469293697 T^{2} + 27292885746930 T^{3} + 1098097521926443092 p T^{4} + 27292885746930 p^{11} T^{5} + 3469293697 p^{22} T^{6} - 9330 p^{33} T^{7} + p^{44} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 75908 T + 370285274849 T^{2} + 95129559794024726 T^{3} + \)\(14\!\cdots\!60\)\( T^{4} + 95129559794024726 p^{11} T^{5} + 370285274849 p^{22} T^{6} + 75908 p^{33} T^{7} + p^{44} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 91106 T + 2076935179360 T^{2} - 966333860499368726 T^{3} + \)\(10\!\cdots\!30\)\( T^{4} - 966333860499368726 p^{11} T^{5} + 2076935179360 p^{22} T^{6} - 91106 p^{33} T^{7} + p^{44} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 4841418 T + 68640912137165 T^{2} + \)\(47\!\cdots\!30\)\( T^{3} + \)\(26\!\cdots\!04\)\( T^{4} + \)\(47\!\cdots\!30\)\( p^{11} T^{5} + 68640912137165 p^{22} T^{6} + 4841418 p^{33} T^{7} + p^{44} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 69280708 T + 5358222589662704 T^{2} - \)\(21\!\cdots\!16\)\( T^{3} + \)\(84\!\cdots\!58\)\( T^{4} - \)\(21\!\cdots\!16\)\( p^{11} T^{5} + 5358222589662704 p^{22} T^{6} - 69280708 p^{33} T^{7} + p^{44} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 148473530 T + 46113450686682152 T^{2} - \)\(45\!\cdots\!26\)\( T^{3} + \)\(81\!\cdots\!46\)\( T^{4} - \)\(45\!\cdots\!26\)\( p^{11} T^{5} + 46113450686682152 p^{22} T^{6} - 148473530 p^{33} T^{7} + p^{44} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 240106042 T + 82567019062564312 T^{2} - \)\(17\!\cdots\!50\)\( T^{3} + \)\(93\!\cdots\!62\)\( p T^{4} - \)\(17\!\cdots\!50\)\( p^{11} T^{5} + 82567019062564312 p^{22} T^{6} - 240106042 p^{33} T^{7} + p^{44} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 540386330 T + 555687730634485588 T^{2} + \)\(29\!\cdots\!10\)\( T^{3} + \)\(13\!\cdots\!50\)\( T^{4} + \)\(29\!\cdots\!10\)\( p^{11} T^{5} + 555687730634485588 p^{22} T^{6} + 540386330 p^{33} T^{7} + p^{44} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 2301171322 T + 3914759672450526884 T^{2} + \)\(42\!\cdots\!06\)\( T^{3} + \)\(91\!\cdots\!86\)\( p T^{4} + \)\(42\!\cdots\!06\)\( p^{11} T^{5} + 3914759672450526884 p^{22} T^{6} + 2301171322 p^{33} T^{7} + p^{44} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 1435224830 T + 1647199421913271585 T^{2} + \)\(22\!\cdots\!10\)\( T^{3} + \)\(26\!\cdots\!48\)\( T^{4} + \)\(22\!\cdots\!10\)\( p^{11} T^{5} + 1647199421913271585 p^{22} T^{6} + 1435224830 p^{33} T^{7} + p^{44} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 23672510 T + 9070763729925374105 T^{2} + \)\(40\!\cdots\!06\)\( T^{3} + \)\(32\!\cdots\!12\)\( T^{4} + \)\(40\!\cdots\!06\)\( p^{11} T^{5} + 9070763729925374105 p^{22} T^{6} + 23672510 p^{33} T^{7} + p^{44} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 157582634 T + 20433670332900792392 T^{2} - \)\(26\!\cdots\!26\)\( T^{3} + \)\(19\!\cdots\!30\)\( T^{4} - \)\(26\!\cdots\!26\)\( p^{11} T^{5} + 20433670332900792392 p^{22} T^{6} + 157582634 p^{33} T^{7} + p^{44} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 114571968 p T + 78062666866974333212 T^{2} + \)\(18\!\cdots\!76\)\( T^{3} + \)\(20\!\cdots\!34\)\( T^{4} + \)\(18\!\cdots\!76\)\( p^{11} T^{5} + 78062666866974333212 p^{22} T^{6} + 114571968 p^{34} T^{7} + p^{44} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 17045196250 T + \)\(25\!\cdots\!13\)\( T^{2} + \)\(23\!\cdots\!42\)\( T^{3} + \)\(18\!\cdots\!16\)\( T^{4} + \)\(23\!\cdots\!42\)\( p^{11} T^{5} + \)\(25\!\cdots\!13\)\( p^{22} T^{6} + 17045196250 p^{33} T^{7} + p^{44} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 38072716248 T + \)\(81\!\cdots\!24\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!22\)\( T^{4} + \)\(13\!\cdots\!20\)\( p^{11} T^{5} + \)\(81\!\cdots\!24\)\( p^{22} T^{6} + 38072716248 p^{33} T^{7} + p^{44} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 35498629184 T + \)\(11\!\cdots\!04\)\( T^{2} + \)\(20\!\cdots\!96\)\( T^{3} + \)\(37\!\cdots\!70\)\( T^{4} + \)\(20\!\cdots\!96\)\( p^{11} T^{5} + \)\(11\!\cdots\!04\)\( p^{22} T^{6} + 35498629184 p^{33} T^{7} + p^{44} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 65896976010 T + \)\(28\!\cdots\!29\)\( T^{2} + \)\(77\!\cdots\!74\)\( T^{3} + \)\(16\!\cdots\!00\)\( T^{4} + \)\(77\!\cdots\!74\)\( p^{11} T^{5} + \)\(28\!\cdots\!29\)\( p^{22} T^{6} + 65896976010 p^{33} T^{7} + p^{44} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 58217838490 T + \)\(26\!\cdots\!00\)\( T^{2} + \)\(84\!\cdots\!82\)\( T^{3} + \)\(27\!\cdots\!50\)\( T^{4} + \)\(84\!\cdots\!82\)\( p^{11} T^{5} + \)\(26\!\cdots\!00\)\( p^{22} T^{6} + 58217838490 p^{33} T^{7} + p^{44} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 19948336098 T + \)\(82\!\cdots\!48\)\( T^{2} - \)\(13\!\cdots\!10\)\( T^{3} + \)\(12\!\cdots\!58\)\( T^{4} - \)\(13\!\cdots\!10\)\( p^{11} T^{5} + \)\(82\!\cdots\!48\)\( p^{22} T^{6} + 19948336098 p^{33} T^{7} + p^{44} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 111132841170 T + \)\(15\!\cdots\!88\)\( T^{2} + \)\(10\!\cdots\!82\)\( T^{3} + \)\(72\!\cdots\!02\)\( T^{4} + \)\(10\!\cdots\!82\)\( p^{11} T^{5} + \)\(15\!\cdots\!88\)\( p^{22} T^{6} + 111132841170 p^{33} T^{7} + p^{44} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 130052309396 T + \)\(18\!\cdots\!32\)\( T^{2} + \)\(10\!\cdots\!64\)\( T^{3} + \)\(11\!\cdots\!74\)\( T^{4} + \)\(10\!\cdots\!64\)\( p^{11} T^{5} + \)\(18\!\cdots\!32\)\( p^{22} T^{6} + 130052309396 p^{33} T^{7} + p^{44} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.442748390249932362792008177724, −7.50571777324897596847881481626, −7.48127824181539440493457290025, −7.47634352195697321877371564873, −7.06853100154932246041344515716, −6.60473170607742447614621376496, −6.32964779455272952454695089844, −6.31897986772350152852273523305, −6.26712902118507956638716374283, −5.54139531052858198232580793459, −5.30536962914289728455878075051, −5.10787080676977799393840528765, −4.80670668596526534037999624869, −4.54186377494946805970727195541, −4.42542692914373653503661056600, −4.09969533923386771058192652932, −4.01747282745851684137552852930, −3.20224952630033381969634179227, −3.11575232883577995821884856345, −2.90995643456514656996636784509, −2.67120622531361789558290851817, −1.64693005622651213980828093328, −1.53616677996292440083068023839, −1.32734223310112118646234691813, −1.30727238624580059118575229369, 0, 0, 0, 0, 1.30727238624580059118575229369, 1.32734223310112118646234691813, 1.53616677996292440083068023839, 1.64693005622651213980828093328, 2.67120622531361789558290851817, 2.90995643456514656996636784509, 3.11575232883577995821884856345, 3.20224952630033381969634179227, 4.01747282745851684137552852930, 4.09969533923386771058192652932, 4.42542692914373653503661056600, 4.54186377494946805970727195541, 4.80670668596526534037999624869, 5.10787080676977799393840528765, 5.30536962914289728455878075051, 5.54139531052858198232580793459, 6.26712902118507956638716374283, 6.31897986772350152852273523305, 6.32964779455272952454695089844, 6.60473170607742447614621376496, 7.06853100154932246041344515716, 7.47634352195697321877371564873, 7.48127824181539440493457290025, 7.50571777324897596847881481626, 8.442748390249932362792008177724