Dirichlet series
| L(s) = 1 | + 128·2-s − 972·3-s + 1.02e4·4-s − 2.86e3·5-s − 1.24e5·6-s − 6.67e4·7-s + 6.55e5·8-s + 5.90e5·9-s − 3.66e5·10-s − 1.49e4·11-s − 9.95e6·12-s + 1.72e6·13-s − 8.54e6·14-s + 2.78e6·15-s + 3.67e7·16-s − 5.67e6·17-s + 7.55e7·18-s + 9.38e6·19-s − 2.93e7·20-s + 6.48e7·21-s − 1.91e6·22-s − 1.19e7·23-s − 6.37e8·24-s − 2.95e7·25-s + 2.20e8·26-s − 2.86e8·27-s − 6.83e8·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 5·4-s − 0.409·5-s − 6.53·6-s − 1.50·7-s + 7.07·8-s + 10/3·9-s − 1.15·10-s − 0.0280·11-s − 11.5·12-s + 1.28·13-s − 4.24·14-s + 0.945·15-s + 35/4·16-s − 0.970·17-s + 9.42·18-s + 0.869·19-s − 2.04·20-s + 3.46·21-s − 0.0793·22-s − 0.388·23-s − 16.3·24-s − 0.604·25-s + 3.63·26-s − 3.84·27-s − 7.50·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 17^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.77243\times 10^{7}\) |
| Root analytic conductor: | \(8.85273\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 17^{4} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )\) |
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 | $C_1$ | \( ( 1 + p^{5} T )^{4} \) | |
| 17 | $C_1$ | \( ( 1 + p^{5} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 2862 T + 7543273 p T^{2} + 4578026958 p^{2} T^{3} - 1638868052448 p^{3} T^{4} + 4578026958 p^{13} T^{5} + 7543273 p^{23} T^{6} + 2862 p^{33} T^{7} + p^{44} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 9532 p T + 6768219616 T^{2} + 301510690073140 T^{3} + 2558615814642862258 p T^{4} + 301510690073140 p^{11} T^{5} + 6768219616 p^{22} T^{6} + 9532 p^{34} T^{7} + p^{44} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 1362 p T + 730529848805 T^{2} + 4256244825924258 p T^{3} + \)\(27\!\cdots\!28\)\( T^{4} + 4256244825924258 p^{12} T^{5} + 730529848805 p^{22} T^{6} + 1362 p^{34} T^{7} + p^{44} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 1721258 T + 2580088684897 T^{2} - 3113385240970838570 T^{3} + \)\(47\!\cdots\!56\)\( T^{4} - 3113385240970838570 p^{11} T^{5} + 2580088684897 p^{22} T^{6} - 1721258 p^{33} T^{7} + p^{44} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 9387998 T + 222084515578549 T^{2} - \)\(32\!\cdots\!42\)\( T^{3} + \)\(24\!\cdots\!20\)\( T^{4} - \)\(32\!\cdots\!42\)\( p^{11} T^{5} + 222084515578549 p^{22} T^{6} - 9387998 p^{33} T^{7} + p^{44} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 11994042 T + 1607124781014185 T^{2} + \)\(10\!\cdots\!22\)\( T^{3} + \)\(19\!\cdots\!32\)\( T^{4} + \)\(10\!\cdots\!22\)\( p^{11} T^{5} + 1607124781014185 p^{22} T^{6} + 11994042 p^{33} T^{7} + p^{44} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 62772048 T + 9360229454275208 T^{2} + \)\(87\!\cdots\!76\)\( T^{3} + \)\(26\!\cdots\!82\)\( T^{4} + \)\(87\!\cdots\!76\)\( p^{11} T^{5} + 9360229454275208 p^{22} T^{6} + 62772048 p^{33} T^{7} + p^{44} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 513497848 T + 172182166311115204 T^{2} + \)\(38\!\cdots\!64\)\( T^{3} + \)\(69\!\cdots\!26\)\( T^{4} + \)\(38\!\cdots\!64\)\( p^{11} T^{5} + 172182166311115204 p^{22} T^{6} + 513497848 p^{33} T^{7} + p^{44} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 406992724 T + 360261400147649860 T^{2} + \)\(80\!\cdots\!32\)\( T^{3} + \)\(63\!\cdots\!58\)\( T^{4} + \)\(80\!\cdots\!32\)\( p^{11} T^{5} + 360261400147649860 p^{22} T^{6} + 406992724 p^{33} T^{7} + p^{44} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 216479550 T + 794507566861609793 T^{2} + \)\(83\!\cdots\!38\)\( T^{3} + \)\(24\!\cdots\!96\)\( T^{4} + \)\(83\!\cdots\!38\)\( p^{11} T^{5} + 794507566861609793 p^{22} T^{6} - 216479550 p^{33} T^{7} + p^{44} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 268964902 T + 3282288230513886133 T^{2} + \)\(75\!\cdots\!42\)\( T^{3} + \)\(43\!\cdots\!64\)\( T^{4} + \)\(75\!\cdots\!42\)\( p^{11} T^{5} + 3282288230513886133 p^{22} T^{6} + 268964902 p^{33} T^{7} + p^{44} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 2493112980 T + 9386307347735478560 T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!18\)\( T^{4} - \)\(17\!\cdots\!40\)\( p^{11} T^{5} + 9386307347735478560 p^{22} T^{6} - 2493112980 p^{33} T^{7} + p^{44} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 2236151016 T + 23073339234745565756 T^{2} + \)\(39\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!42\)\( T^{4} + \)\(39\!\cdots\!00\)\( p^{11} T^{5} + 23073339234745565756 p^{22} T^{6} + 2236151016 p^{33} T^{7} + p^{44} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 17008682940 T + \)\(18\!\cdots\!72\)\( T^{2} + \)\(13\!\cdots\!40\)\( T^{3} + \)\(85\!\cdots\!34\)\( T^{4} + \)\(13\!\cdots\!40\)\( p^{11} T^{5} + \)\(18\!\cdots\!72\)\( p^{22} T^{6} + 17008682940 p^{33} T^{7} + p^{44} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 21397284628 T + \)\(31\!\cdots\!00\)\( T^{2} + \)\(31\!\cdots\!56\)\( T^{3} + \)\(23\!\cdots\!86\)\( T^{4} + \)\(31\!\cdots\!56\)\( p^{11} T^{5} + \)\(31\!\cdots\!00\)\( p^{22} T^{6} + 21397284628 p^{33} T^{7} + p^{44} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 9294323200 T + \)\(26\!\cdots\!88\)\( T^{2} + \)\(23\!\cdots\!44\)\( T^{3} + \)\(41\!\cdots\!02\)\( T^{4} + \)\(23\!\cdots\!44\)\( p^{11} T^{5} + \)\(26\!\cdots\!88\)\( p^{22} T^{6} + 9294323200 p^{33} T^{7} + p^{44} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 63776133180 T + \)\(23\!\cdots\!52\)\( T^{2} + \)\(57\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!58\)\( T^{4} + \)\(57\!\cdots\!60\)\( p^{11} T^{5} + \)\(23\!\cdots\!52\)\( p^{22} T^{6} + 63776133180 p^{33} T^{7} + p^{44} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 30041305648 T + \)\(90\!\cdots\!84\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!06\)\( T^{4} + \)\(11\!\cdots\!20\)\( p^{11} T^{5} + \)\(90\!\cdots\!84\)\( p^{22} T^{6} + 30041305648 p^{33} T^{7} + p^{44} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 49690430992 T + \)\(35\!\cdots\!32\)\( T^{2} + \)\(10\!\cdots\!56\)\( T^{3} + \)\(40\!\cdots\!34\)\( T^{4} + \)\(10\!\cdots\!56\)\( p^{11} T^{5} + \)\(35\!\cdots\!32\)\( p^{22} T^{6} + 49690430992 p^{33} T^{7} + p^{44} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 75290662044 T + \)\(68\!\cdots\!36\)\( T^{2} + \)\(30\!\cdots\!64\)\( T^{3} + \)\(14\!\cdots\!46\)\( T^{4} + \)\(30\!\cdots\!64\)\( p^{11} T^{5} + \)\(68\!\cdots\!36\)\( p^{22} T^{6} + 75290662044 p^{33} T^{7} + p^{44} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 52786810860 T + \)\(87\!\cdots\!28\)\( T^{2} + \)\(35\!\cdots\!12\)\( T^{3} + \)\(35\!\cdots\!42\)\( T^{4} + \)\(35\!\cdots\!12\)\( p^{11} T^{5} + \)\(87\!\cdots\!28\)\( p^{22} T^{6} + 52786810860 p^{33} T^{7} + p^{44} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 172006271332 T + \)\(27\!\cdots\!80\)\( T^{2} + \)\(31\!\cdots\!52\)\( T^{3} + \)\(29\!\cdots\!22\)\( T^{4} + \)\(31\!\cdots\!52\)\( p^{11} T^{5} + \)\(27\!\cdots\!80\)\( p^{22} T^{6} + 172006271332 p^{33} T^{7} + p^{44} T^{8} \) | |
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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
−8.555197631585554340763176342965, −7.72343998227355546687389773778, −7.47021056430969957566286262234, −7.45067769285842368474024001124, −7.36895095642144055131154797580, −6.63974852582092372137833810892, −6.52470838044254883071452362873, −6.36634903730265086866382819161, −6.02937406509029092683678909758, −5.77349513580597165342895102544, −5.64449116278735978511945266997, −5.37156054462880920855419124934, −5.11964315743351442037859848363, −4.42539085317158072532826315105, −4.35071678893401498534818804919, −4.28124883759910503987755068894, −3.89806135147271382556594488522, −3.36882424561531910108385443869, −3.25750726609204082391567300461, −2.95905017465603506715773378756, −2.65635903620859309620781410331, −1.79200425713018877019389352796, −1.55250784744209176053243533360, −1.38551713645839390915216363663, −1.34042510440461748474203618202, 0, 0, 0, 0, 1.34042510440461748474203618202, 1.38551713645839390915216363663, 1.55250784744209176053243533360, 1.79200425713018877019389352796, 2.65635903620859309620781410331, 2.95905017465603506715773378756, 3.25750726609204082391567300461, 3.36882424561531910108385443869, 3.89806135147271382556594488522, 4.28124883759910503987755068894, 4.35071678893401498534818804919, 4.42539085317158072532826315105, 5.11964315743351442037859848363, 5.37156054462880920855419124934, 5.64449116278735978511945266997, 5.77349513580597165342895102544, 6.02937406509029092683678909758, 6.36634903730265086866382819161, 6.52470838044254883071452362873, 6.63974852582092372137833810892, 7.36895095642144055131154797580, 7.45067769285842368474024001124, 7.47021056430969957566286262234, 7.72343998227355546687389773778, 8.555197631585554340763176342965