Dirichlet series
| L(s) = 1 | − 64·2-s + 324·3-s + 2.56e3·4-s + 2.50e3·5-s − 2.07e4·6-s − 2.31e3·7-s − 8.19e4·8-s + 6.56e4·9-s − 1.60e5·10-s − 1.05e4·11-s + 8.29e5·12-s − 1.14e5·13-s + 1.48e5·14-s + 8.10e5·15-s + 2.29e6·16-s − 3.86e5·17-s − 4.19e6·18-s − 2.28e5·19-s + 6.40e6·20-s − 7.51e5·21-s + 6.72e5·22-s + 3.61e5·23-s − 2.65e7·24-s + 3.90e6·25-s + 7.31e6·26-s + 1.06e7·27-s − 5.93e6·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 5·4-s + 1.78·5-s − 6.53·6-s − 0.364·7-s − 7.07·8-s + 10/3·9-s − 5.05·10-s − 0.216·11-s + 11.5·12-s − 1.10·13-s + 1.03·14-s + 4.13·15-s + 35/4·16-s − 1.12·17-s − 9.42·18-s − 0.402·19-s + 8.94·20-s − 0.842·21-s + 0.612·22-s + 0.269·23-s − 16.3·24-s + 2·25-s + 3.13·26-s + 3.84·27-s − 1.82·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.62782\times 10^{9}\) |
| Root analytic conductor: | \(14.1726\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | 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 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| 5 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| good | 7 | $C_2 \wr S_4$ | \( 1 + 2318 T + 56153569 T^{2} + 55391322874 p T^{3} + 32395720546232 p^{2} T^{4} + 55391322874 p^{10} T^{5} + 56153569 p^{18} T^{6} + 2318 p^{27} T^{7} + p^{36} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 10508 T + 4067423441 T^{2} - 36906349231022 T^{3} + 10970663978186276924 T^{4} - 36906349231022 p^{9} T^{5} + 4067423441 p^{18} T^{6} + 10508 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 386792 T + 418201943561 T^{2} + 118109372452322846 T^{3} + \)\(72\!\cdots\!76\)\( T^{4} + 118109372452322846 p^{9} T^{5} + 418201943561 p^{18} T^{6} + 386792 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 228476 T + 59285709316 p T^{2} + 210216222850684716 T^{3} + \)\(51\!\cdots\!66\)\( T^{4} + 210216222850684716 p^{9} T^{5} + 59285709316 p^{19} T^{6} + 228476 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 361088 T + 2517829962113 T^{2} - 4275984554069659554 T^{3} + \)\(36\!\cdots\!48\)\( T^{4} - 4275984554069659554 p^{9} T^{5} + 2517829962113 p^{18} T^{6} - 361088 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 1053374 T + 31726389362480 T^{2} - 83109881526263658202 T^{3} + \)\(49\!\cdots\!26\)\( T^{4} - 83109881526263658202 p^{9} T^{5} + 31726389362480 p^{18} T^{6} - 1053374 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 12118 T + 56525659977760 T^{2} - \)\(12\!\cdots\!86\)\( T^{3} + \)\(15\!\cdots\!06\)\( T^{4} - \)\(12\!\cdots\!86\)\( p^{9} T^{5} + 56525659977760 p^{18} T^{6} - 12118 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 983622 T + 143456717491849 T^{2} - \)\(38\!\cdots\!02\)\( T^{3} + \)\(35\!\cdots\!88\)\( T^{4} - \)\(38\!\cdots\!02\)\( p^{9} T^{5} + 143456717491849 p^{18} T^{6} - 983622 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 5900234 T + 1315110053342465 T^{2} + \)\(57\!\cdots\!66\)\( T^{3} + \)\(64\!\cdots\!44\)\( T^{4} + \)\(57\!\cdots\!66\)\( p^{9} T^{5} + 1315110053342465 p^{18} T^{6} + 5900234 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 7525148 T + 614474905369180 T^{2} + \)\(41\!\cdots\!64\)\( p T^{3} + \)\(29\!\cdots\!30\)\( T^{4} + \)\(41\!\cdots\!64\)\( p^{10} T^{5} + 614474905369180 p^{18} T^{6} + 7525148 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 52858086 T + 5338702005083480 T^{2} - \)\(18\!\cdots\!34\)\( T^{3} + \)\(94\!\cdots\!62\)\( T^{4} - \)\(18\!\cdots\!34\)\( p^{9} T^{5} + 5338702005083480 p^{18} T^{6} - 52858086 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 39944454 T + 9265475525829857 T^{2} + \)\(44\!\cdots\!46\)\( T^{3} + \)\(39\!\cdots\!84\)\( T^{4} + \)\(44\!\cdots\!46\)\( p^{9} T^{5} + 9265475525829857 p^{18} T^{6} + 39944454 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 221262654 T + 47227770124652420 T^{2} + \)\(57\!\cdots\!14\)\( T^{3} + \)\(65\!\cdots\!78\)\( T^{4} + \)\(57\!\cdots\!14\)\( p^{9} T^{5} + 47227770124652420 p^{18} T^{6} + 221262654 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 301613618 T + 77415563909053921 T^{2} + \)\(11\!\cdots\!54\)\( T^{3} + \)\(15\!\cdots\!60\)\( T^{4} + \)\(11\!\cdots\!54\)\( p^{9} T^{5} + 77415563909053921 p^{18} T^{6} + 301613618 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 107925130 T + 77586145850746972 T^{2} - \)\(62\!\cdots\!62\)\( T^{3} + \)\(29\!\cdots\!02\)\( T^{4} - \)\(62\!\cdots\!62\)\( p^{9} T^{5} + 77586145850746972 p^{18} T^{6} - 107925130 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 407933556 T + 164059652846232713 T^{2} + \)\(31\!\cdots\!34\)\( T^{3} + \)\(84\!\cdots\!12\)\( T^{4} + \)\(31\!\cdots\!34\)\( p^{9} T^{5} + 164059652846232713 p^{18} T^{6} + 407933556 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 649353258 T + 320414750305756864 T^{2} + \)\(99\!\cdots\!82\)\( T^{3} + \)\(28\!\cdots\!26\)\( T^{4} + \)\(99\!\cdots\!82\)\( p^{9} T^{5} + 320414750305756864 p^{18} T^{6} + 649353258 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 1126263686 T + 820505014421459749 T^{2} + \)\(39\!\cdots\!86\)\( T^{3} + \)\(15\!\cdots\!76\)\( T^{4} + \)\(39\!\cdots\!86\)\( p^{9} T^{5} + 820505014421459749 p^{18} T^{6} + 1126263686 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 334595286 T + 500170612609827956 T^{2} + \)\(10\!\cdots\!30\)\( T^{3} + \)\(11\!\cdots\!26\)\( T^{4} + \)\(10\!\cdots\!30\)\( p^{9} T^{5} + 500170612609827956 p^{18} T^{6} + 334595286 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 110458574 T + 900529893969203225 T^{2} - \)\(16\!\cdots\!38\)\( T^{3} + \)\(39\!\cdots\!88\)\( T^{4} - \)\(16\!\cdots\!38\)\( p^{9} T^{5} + 900529893969203225 p^{18} T^{6} - 110458574 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 1740509692 T + 2594422465595203009 T^{2} - \)\(23\!\cdots\!50\)\( T^{3} + \)\(23\!\cdots\!36\)\( T^{4} - \)\(23\!\cdots\!50\)\( p^{9} T^{5} + 2594422465595203009 p^{18} T^{6} - 1740509692 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−7.50254240162745471597782006813, −6.91813781188452584118229645415, −6.74245715372512877483875416438, −6.72232603628520017946959443805, −6.54722777533232701406749694036, −6.01223569641945016346645923266, −5.80595770550518308422320160213, −5.62985451343871322939938693494, −5.42722696520941275324366383477, −4.67297118527955522414450304264, −4.53543538587412433334876245871, −4.31836367514455532010064209140, −4.28768708961747200488649612708, −3.29384689940302470940943963269, −3.09304406043448495104648128920, −3.08505289291431389210217858985, −3.07416059664464844384721396600, −2.45862973258161884786341342761, −2.25512968082336090468810871186, −2.11989782465686886574078453204, −2.08376777095652564214493632048, −1.38430454134213396285699183348, −1.34089381600654668200439514601, −1.26915474087748008324287986829, −1.08428842447814255901725301273, 0, 0, 0, 0, 1.08428842447814255901725301273, 1.26915474087748008324287986829, 1.34089381600654668200439514601, 1.38430454134213396285699183348, 2.08376777095652564214493632048, 2.11989782465686886574078453204, 2.25512968082336090468810871186, 2.45862973258161884786341342761, 3.07416059664464844384721396600, 3.08505289291431389210217858985, 3.09304406043448495104648128920, 3.29384689940302470940943963269, 4.28768708961747200488649612708, 4.31836367514455532010064209140, 4.53543538587412433334876245871, 4.67297118527955522414450304264, 5.42722696520941275324366383477, 5.62985451343871322939938693494, 5.80595770550518308422320160213, 6.01223569641945016346645923266, 6.54722777533232701406749694036, 6.72232603628520017946959443805, 6.74245715372512877483875416438, 6.91813781188452584118229645415, 7.50254240162745471597782006813