Dirichlet series
| L(s) = 1 | − 2.04e3·2-s − 7.87e4·3-s + 2.62e6·4-s − 4.26e5·5-s + 1.61e8·6-s + 1.01e8·7-s − 2.68e9·8-s + 3.87e9·9-s + 8.72e8·10-s + 5.44e9·11-s − 2.06e11·12-s + 4.24e10·13-s − 2.08e11·14-s + 3.35e10·15-s + 2.40e12·16-s − 2.09e11·17-s − 7.93e12·18-s − 7.77e11·19-s − 1.11e12·20-s − 8.01e12·21-s − 1.11e13·22-s + 3.67e12·23-s + 2.11e14·24-s − 3.95e13·25-s − 8.68e13·26-s − 1.52e14·27-s + 2.66e14·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s − 0.0975·5-s + 6.53·6-s + 0.953·7-s − 7.07·8-s + 10/3·9-s + 0.276·10-s + 0.695·11-s − 11.5·12-s + 1.10·13-s − 2.69·14-s + 0.225·15-s + 35/4·16-s − 0.428·17-s − 9.42·18-s − 0.553·19-s − 0.487·20-s − 2.20·21-s − 1.96·22-s + 0.424·23-s + 16.3·24-s − 2.07·25-s − 3.13·26-s − 3.84·27-s + 4.76·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+19/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(37015056\) = \(2^{4} \cdot 3^{4} \cdot 13^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.01468\times 10^{9}\) |
| Root analytic conductor: | \(13.3595\) |
| Motivic weight: | \(19\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 37015056,\ (\ :19/2, 19/2, 19/2, 19/2),\ 1)\) |
Particular Values
| \(L(10)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{21}{2})\) | 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 | $C_1$ | \( ( 1 + p^{9} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + p^{9} T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 - p^{9} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 85238 p T + 1588579709496 p^{2} T^{2} + 438918947157527346 p^{3} T^{3} + \)\(15\!\cdots\!54\)\( p^{4} T^{4} + 438918947157527346 p^{22} T^{5} + 1588579709496 p^{40} T^{6} + 85238 p^{58} T^{7} + p^{76} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 101834718 T + 6057724014132092 p T^{2} - \)\(67\!\cdots\!34\)\( p^{2} T^{3} + \)\(29\!\cdots\!50\)\( p^{4} T^{4} - \)\(67\!\cdots\!34\)\( p^{21} T^{5} + 6057724014132092 p^{39} T^{6} - 101834718 p^{57} T^{7} + p^{76} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 494559856 p T + \)\(13\!\cdots\!72\)\( T^{2} - \)\(49\!\cdots\!20\)\( p T^{3} + \)\(83\!\cdots\!06\)\( p^{2} T^{4} - \)\(49\!\cdots\!20\)\( p^{20} T^{5} + \)\(13\!\cdots\!72\)\( p^{38} T^{6} - 494559856 p^{58} T^{7} + p^{76} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 209603969800 T + \)\(30\!\cdots\!00\)\( p T^{2} - \)\(15\!\cdots\!00\)\( p^{2} T^{3} + \)\(22\!\cdots\!86\)\( p^{3} T^{4} - \)\(15\!\cdots\!00\)\( p^{21} T^{5} + \)\(30\!\cdots\!00\)\( p^{39} T^{6} + 209603969800 p^{57} T^{7} + p^{76} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 777897302166 T + \)\(20\!\cdots\!56\)\( p T^{2} + \)\(32\!\cdots\!14\)\( T^{3} + \)\(11\!\cdots\!94\)\( T^{4} + \)\(32\!\cdots\!14\)\( p^{19} T^{5} + \)\(20\!\cdots\!56\)\( p^{39} T^{6} + 777897302166 p^{57} T^{7} + p^{76} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 3670753362720 T + \)\(25\!\cdots\!48\)\( T^{2} - \)\(73\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!14\)\( T^{4} - \)\(73\!\cdots\!20\)\( p^{19} T^{5} + \)\(25\!\cdots\!48\)\( p^{38} T^{6} - 3670753362720 p^{57} T^{7} + p^{76} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 4809047780100 T + \)\(88\!\cdots\!84\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(80\!\cdots\!86\)\( T^{4} + \)\(19\!\cdots\!00\)\( p^{19} T^{5} + \)\(88\!\cdots\!84\)\( p^{38} T^{6} + 4809047780100 p^{57} T^{7} + p^{76} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 248970233209042 T + \)\(20\!\cdots\!36\)\( p T^{2} + \)\(91\!\cdots\!82\)\( T^{3} + \)\(17\!\cdots\!54\)\( T^{4} + \)\(91\!\cdots\!82\)\( p^{19} T^{5} + \)\(20\!\cdots\!36\)\( p^{39} T^{6} + 248970233209042 p^{57} T^{7} + p^{76} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 1007898069211452 T + \)\(24\!\cdots\!04\)\( T^{2} + \)\(18\!\cdots\!24\)\( T^{3} + \)\(22\!\cdots\!50\)\( T^{4} + \)\(18\!\cdots\!24\)\( p^{19} T^{5} + \)\(24\!\cdots\!04\)\( p^{38} T^{6} + 1007898069211452 p^{57} T^{7} + p^{76} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 2296930311264622 T + \)\(14\!\cdots\!40\)\( T^{2} + \)\(23\!\cdots\!26\)\( T^{3} + \)\(21\!\cdots\!18\)\( p T^{4} + \)\(23\!\cdots\!26\)\( p^{19} T^{5} + \)\(14\!\cdots\!40\)\( p^{38} T^{6} + 2296930311264622 p^{57} T^{7} + p^{76} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 4152180081684480 T + \)\(25\!\cdots\!00\)\( T^{2} + \)\(93\!\cdots\!60\)\( T^{3} + \)\(43\!\cdots\!98\)\( T^{4} + \)\(93\!\cdots\!60\)\( p^{19} T^{5} + \)\(25\!\cdots\!00\)\( p^{38} T^{6} + 4152180081684480 p^{57} T^{7} + p^{76} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 4883972743480516 T + \)\(19\!\cdots\!28\)\( T^{2} + \)\(82\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!26\)\( T^{4} + \)\(82\!\cdots\!40\)\( p^{19} T^{5} + \)\(19\!\cdots\!28\)\( p^{38} T^{6} + 4883972743480516 p^{57} T^{7} + p^{76} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 40564903764762008 T + \)\(79\!\cdots\!04\)\( T^{2} - \)\(20\!\cdots\!88\)\( T^{3} - \)\(52\!\cdots\!42\)\( T^{4} - \)\(20\!\cdots\!88\)\( p^{19} T^{5} + \)\(79\!\cdots\!04\)\( p^{38} T^{6} - 40564903764762008 p^{57} T^{7} + p^{76} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 14402074800094012 T + \)\(10\!\cdots\!32\)\( T^{2} - \)\(16\!\cdots\!44\)\( T^{3} + \)\(63\!\cdots\!54\)\( T^{4} - \)\(16\!\cdots\!44\)\( p^{19} T^{5} + \)\(10\!\cdots\!32\)\( p^{38} T^{6} - 14402074800094012 p^{57} T^{7} + p^{76} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 8786189989864188 T + \)\(11\!\cdots\!16\)\( T^{2} - \)\(27\!\cdots\!28\)\( T^{3} + \)\(13\!\cdots\!14\)\( T^{4} - \)\(27\!\cdots\!28\)\( p^{19} T^{5} + \)\(11\!\cdots\!16\)\( p^{38} T^{6} - 8786189989864188 p^{57} T^{7} + p^{76} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 387443032289073470 T + \)\(14\!\cdots\!64\)\( T^{2} - \)\(23\!\cdots\!50\)\( T^{3} + \)\(61\!\cdots\!42\)\( T^{4} - \)\(23\!\cdots\!50\)\( p^{19} T^{5} + \)\(14\!\cdots\!64\)\( p^{38} T^{6} - 387443032289073470 p^{57} T^{7} + p^{76} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 137902462656716296 T + \)\(42\!\cdots\!12\)\( T^{2} - \)\(70\!\cdots\!60\)\( T^{3} + \)\(83\!\cdots\!86\)\( T^{4} - \)\(70\!\cdots\!60\)\( p^{19} T^{5} + \)\(42\!\cdots\!12\)\( p^{38} T^{6} - 137902462656716296 p^{57} T^{7} + p^{76} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 204315051380208044 T + \)\(93\!\cdots\!24\)\( T^{2} - \)\(14\!\cdots\!08\)\( T^{3} + \)\(34\!\cdots\!54\)\( T^{4} - \)\(14\!\cdots\!08\)\( p^{19} T^{5} + \)\(93\!\cdots\!24\)\( p^{38} T^{6} - 204315051380208044 p^{57} T^{7} + p^{76} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 71498410696538840 T + \)\(22\!\cdots\!08\)\( T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(31\!\cdots\!38\)\( T^{4} - \)\(13\!\cdots\!20\)\( p^{19} T^{5} + \)\(22\!\cdots\!08\)\( p^{38} T^{6} - 71498410696538840 p^{57} T^{7} + p^{76} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 4535420367809710620 T + \)\(15\!\cdots\!16\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(74\!\cdots\!82\)\( T^{4} - \)\(38\!\cdots\!00\)\( p^{19} T^{5} + \)\(15\!\cdots\!16\)\( p^{38} T^{6} - 4535420367809710620 p^{57} T^{7} + p^{76} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 8531436985684203230 T + \)\(63\!\cdots\!24\)\( T^{2} - \)\(28\!\cdots\!30\)\( T^{3} + \)\(11\!\cdots\!06\)\( T^{4} - \)\(28\!\cdots\!30\)\( p^{19} T^{5} + \)\(63\!\cdots\!24\)\( p^{38} T^{6} - 8531436985684203230 p^{57} T^{7} + p^{76} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 16286061647499632172 T + \)\(31\!\cdots\!08\)\( T^{2} - \)\(29\!\cdots\!08\)\( T^{3} + \)\(29\!\cdots\!50\)\( T^{4} - \)\(29\!\cdots\!08\)\( p^{19} T^{5} + \)\(31\!\cdots\!08\)\( p^{38} T^{6} - 16286061647499632172 p^{57} T^{7} + p^{76} T^{8} \) | |
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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
−8.059647836531085983159993475847, −7.24248778654692178752490524456, −7.17426343103758119024327694565, −7.10448082893840874940786290383, −6.78525021091593468025440228308, −6.21830824463452252671864804464, −6.21713918535578218569836654617, −6.14352673076581068874289801041, −5.73845349354204198318344381492, −5.27488062864571557325897590682, −5.03102251447807373088758528113, −4.76343239086742227239162791261, −4.65601784892103536362212369593, −3.72082148062572712430730439619, −3.67799017728169551872928692995, −3.46678671997404946072787904419, −3.41863901114309876788537838866, −2.26827365043761909860451113293, −2.06736917483966353237802387526, −2.02282254299631457677133338863, −1.94900116508452943261451373043, −1.20146099234968972311859818631, −1.16364337317562793994411206830, −1.15645543594732922157170749972, −0.74926262769495495807827889909, 0, 0, 0, 0, 0.74926262769495495807827889909, 1.15645543594732922157170749972, 1.16364337317562793994411206830, 1.20146099234968972311859818631, 1.94900116508452943261451373043, 2.02282254299631457677133338863, 2.06736917483966353237802387526, 2.26827365043761909860451113293, 3.41863901114309876788537838866, 3.46678671997404946072787904419, 3.67799017728169551872928692995, 3.72082148062572712430730439619, 4.65601784892103536362212369593, 4.76343239086742227239162791261, 5.03102251447807373088758528113, 5.27488062864571557325897590682, 5.73845349354204198318344381492, 6.14352673076581068874289801041, 6.21713918535578218569836654617, 6.21830824463452252671864804464, 6.78525021091593468025440228308, 7.10448082893840874940786290383, 7.17426343103758119024327694565, 7.24248778654692178752490524456, 8.059647836531085983159993475847