Dirichlet series
| L(s) = 1 | − 5·2-s − 537·4-s + 2.50e3·5-s − 9.60e3·7-s + 991·8-s − 1.25e4·10-s − 6.45e4·11-s − 2.93e4·13-s + 4.80e4·14-s + 1.69e5·16-s − 2.78e5·17-s − 9.29e5·19-s − 1.34e6·20-s + 3.22e5·22-s + 5.26e5·23-s + 3.90e6·25-s + 1.46e5·26-s + 5.15e6·28-s − 3.65e5·29-s − 4.97e6·31-s − 5.47e5·32-s + 1.39e6·34-s − 2.40e7·35-s + 2.28e7·37-s + 4.64e6·38-s + 2.47e6·40-s + 2.82e7·41-s + ⋯ |
| L(s) = 1 | − 0.220·2-s − 1.04·4-s + 1.78·5-s − 1.51·7-s + 0.0855·8-s − 0.395·10-s − 1.32·11-s − 0.285·13-s + 0.334·14-s + 0.648·16-s − 0.809·17-s − 1.63·19-s − 1.87·20-s + 0.293·22-s + 0.391·23-s + 2·25-s + 0.0630·26-s + 1.58·28-s − 0.0960·29-s − 0.968·31-s − 0.0922·32-s + 0.178·34-s − 2.70·35-s + 2.00·37-s + 0.361·38-s + 0.153·40-s + 1.56·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(3^{8} \cdot 5^{4} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.92774\times 10^{8}\) |
| Root analytic conductor: | \(12.7372\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 3^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
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 | 3 | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 + 5 T + 281 p T^{2} + 563 p^{3} T^{3} + 1167 p^{7} T^{4} + 563 p^{12} T^{5} + 281 p^{19} T^{6} + 5 p^{27} T^{7} + p^{36} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 64546 T + 7269451996 T^{2} + 408023915576066 T^{3} + 24310519363699903350 T^{4} + 408023915576066 p^{9} T^{5} + 7269451996 p^{18} T^{6} + 64546 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 29390 T + 22351274472 T^{2} + 265168323158506 T^{3} + \)\(25\!\cdots\!62\)\( T^{4} + 265168323158506 p^{9} T^{5} + 22351274472 p^{18} T^{6} + 29390 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 278788 T + 473329598692 T^{2} + 95909056587305276 T^{3} + \)\(49\!\cdots\!94\)\( p T^{4} + 95909056587305276 p^{9} T^{5} + 473329598692 p^{18} T^{6} + 278788 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 929142 T + 965216188644 T^{2} + 622557929988811782 T^{3} + \)\(42\!\cdots\!30\)\( T^{4} + 622557929988811782 p^{9} T^{5} + 965216188644 p^{18} T^{6} + 929142 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 526064 T + 2170511578012 T^{2} - 1872889874671841968 T^{3} + \)\(72\!\cdots\!62\)\( T^{4} - 1872889874671841968 p^{9} T^{5} + 2170511578012 p^{18} T^{6} - 526064 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 365860 T + 37670299711220 T^{2} + 31056883269757829164 T^{3} + \)\(67\!\cdots\!98\)\( T^{4} + 31056883269757829164 p^{9} T^{5} + 37670299711220 p^{18} T^{6} + 365860 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 4977954 T + 88072246168420 T^{2} + \)\(37\!\cdots\!34\)\( T^{3} + \)\(32\!\cdots\!62\)\( T^{4} + \)\(37\!\cdots\!34\)\( p^{9} T^{5} + 88072246168420 p^{18} T^{6} + 4977954 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 22846008 T + 605448015384460 T^{2} - \)\(79\!\cdots\!32\)\( T^{3} + \)\(11\!\cdots\!90\)\( T^{4} - \)\(79\!\cdots\!32\)\( p^{9} T^{5} + 605448015384460 p^{18} T^{6} - 22846008 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 28257844 T + 1472108304294052 T^{2} - \)\(27\!\cdots\!32\)\( T^{3} + \)\(74\!\cdots\!02\)\( T^{4} - \)\(27\!\cdots\!32\)\( p^{9} T^{5} + 1472108304294052 p^{18} T^{6} - 28257844 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 23603420 T + 1942778777119212 T^{2} - \)\(34\!\cdots\!64\)\( T^{3} + \)\(14\!\cdots\!22\)\( T^{4} - \)\(34\!\cdots\!64\)\( p^{9} T^{5} + 1942778777119212 p^{18} T^{6} - 23603420 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 30058700 T + 3867120056326412 T^{2} - \)\(90\!\cdots\!72\)\( T^{3} + \)\(62\!\cdots\!94\)\( T^{4} - \)\(90\!\cdots\!72\)\( p^{9} T^{5} + 3867120056326412 p^{18} T^{6} - 30058700 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 113767294 T + 14444775155815384 T^{2} + \)\(90\!\cdots\!30\)\( T^{3} + \)\(67\!\cdots\!46\)\( T^{4} + \)\(90\!\cdots\!30\)\( p^{9} T^{5} + 14444775155815384 p^{18} T^{6} + 113767294 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 151786220 T + 26874891828787196 T^{2} - \)\(32\!\cdots\!56\)\( T^{3} + \)\(34\!\cdots\!46\)\( T^{4} - \)\(32\!\cdots\!56\)\( p^{9} T^{5} + 26874891828787196 p^{18} T^{6} - 151786220 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 191130108 T + 35831813595938644 T^{2} + \)\(51\!\cdots\!32\)\( T^{3} + \)\(54\!\cdots\!30\)\( T^{4} + \)\(51\!\cdots\!32\)\( p^{9} T^{5} + 35831813595938644 p^{18} T^{6} + 191130108 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 147812356 T + 86585970985974924 T^{2} + \)\(10\!\cdots\!16\)\( T^{3} + \)\(33\!\cdots\!50\)\( T^{4} + \)\(10\!\cdots\!16\)\( p^{9} T^{5} + 86585970985974924 p^{18} T^{6} + 147812356 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 37100486 T + 169585368015908684 T^{2} - \)\(56\!\cdots\!94\)\( T^{3} + \)\(15\!\cdots\!50\)\( p T^{4} - \)\(56\!\cdots\!94\)\( p^{9} T^{5} + 169585368015908684 p^{18} T^{6} - 37100486 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 163444574 T + 198269192296166088 T^{2} - \)\(25\!\cdots\!06\)\( T^{3} + \)\(16\!\cdots\!62\)\( T^{4} - \)\(25\!\cdots\!06\)\( p^{9} T^{5} + 198269192296166088 p^{18} T^{6} - 163444574 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 327433200 T + 359528071713938556 T^{2} + \)\(94\!\cdots\!08\)\( T^{3} + \)\(62\!\cdots\!86\)\( T^{4} + \)\(94\!\cdots\!08\)\( p^{9} T^{5} + 359528071713938556 p^{18} T^{6} + 327433200 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 216775352 T + 15468188990873420 T^{2} + \)\(41\!\cdots\!16\)\( T^{3} + \)\(73\!\cdots\!98\)\( T^{4} + \)\(41\!\cdots\!16\)\( p^{9} T^{5} + 15468188990873420 p^{18} T^{6} + 216775352 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 792987912 T + 1284349731754908604 T^{2} - \)\(68\!\cdots\!36\)\( T^{3} + \)\(64\!\cdots\!90\)\( T^{4} - \)\(68\!\cdots\!36\)\( p^{9} T^{5} + 1284349731754908604 p^{18} T^{6} - 792987912 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 640730334 T + 1921696179196466736 T^{2} + \)\(80\!\cdots\!78\)\( T^{3} + \)\(18\!\cdots\!98\)\( T^{4} + \)\(80\!\cdots\!78\)\( p^{9} T^{5} + 1921696179196466736 p^{18} T^{6} + 640730334 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−7.32286952252166934542356455881, −7.19455851909712716526031228806, −6.78538779704046950693374848133, −6.53272801875708598740682084733, −6.22881664407345681420619445058, −6.10960683413165327726538418903, −5.83007043394838359157805641403, −5.66810737170310285334505618964, −5.64228315898845193410868676009, −4.87290866606344729493325032800, −4.78215578802665757611940850140, −4.65058392081981738471723867619, −4.51572843527356287686198526505, −3.84105730711281393251247700434, −3.71970605952549733346946210914, −3.57487286213784815634382578955, −2.96229214630390096444131852386, −2.66836263546744812821894502860, −2.57363592073880507451737530294, −2.34714115200587533640154134117, −2.16340944345734252717458307741, −1.76216374968790785481751072595, −1.13954627883648156359915491816, −1.00762893299744498564416093512, −0.959118277880721762417210956129, 0, 0, 0, 0, 0.959118277880721762417210956129, 1.00762893299744498564416093512, 1.13954627883648156359915491816, 1.76216374968790785481751072595, 2.16340944345734252717458307741, 2.34714115200587533640154134117, 2.57363592073880507451737530294, 2.66836263546744812821894502860, 2.96229214630390096444131852386, 3.57487286213784815634382578955, 3.71970605952549733346946210914, 3.84105730711281393251247700434, 4.51572843527356287686198526505, 4.65058392081981738471723867619, 4.78215578802665757611940850140, 4.87290866606344729493325032800, 5.64228315898845193410868676009, 5.66810737170310285334505618964, 5.83007043394838359157805641403, 6.10960683413165327726538418903, 6.22881664407345681420619445058, 6.53272801875708598740682084733, 6.78538779704046950693374848133, 7.19455851909712716526031228806, 7.32286952252166934542356455881