Dirichlet series
| L(s) = 1 | − 15·2-s − 987·4-s − 1.80e3·5-s + 1.00e4·7-s + 1.07e4·8-s + 2.70e4·10-s − 1.21e5·11-s + 1.42e5·13-s − 1.51e5·14-s + 4.33e5·16-s + 4.95e5·17-s − 8.40e5·19-s + 1.77e6·20-s + 1.82e6·22-s + 5.92e5·23-s − 2.42e6·25-s − 2.14e6·26-s − 9.96e6·28-s − 1.06e7·29-s + 1.28e7·31-s + 2.51e5·32-s − 7.43e6·34-s − 1.82e7·35-s + 7.17e6·37-s + 1.26e7·38-s − 1.94e7·40-s − 9.29e6·41-s + ⋯ |
| L(s) = 1 | − 0.662·2-s − 1.92·4-s − 1.29·5-s + 1.58·7-s + 0.929·8-s + 0.855·10-s − 2.50·11-s + 1.38·13-s − 1.05·14-s + 1.65·16-s + 1.43·17-s − 1.48·19-s + 2.48·20-s + 1.66·22-s + 0.441·23-s − 1.24·25-s − 0.919·26-s − 3.06·28-s − 2.80·29-s + 2.50·31-s + 0.0424·32-s − 0.954·34-s − 2.05·35-s + 0.629·37-s + 0.981·38-s − 1.19·40-s − 0.513·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(10\) |
| Conductor: | \(3^{10} \cdot 13^{5}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(7.94541\times 10^{8}\) |
| Root analytic conductor: | \(7.76267\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(5\) |
| Selberg data: | \((10,\ 3^{10} \cdot 13^{5} ,\ ( \ : 9/2, 9/2, 9/2, 9/2, 9/2 ),\ -1 )\) |
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 | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 - p^{4} T )^{5} \) | |
| good | 2 | $C_2 \wr S_5$ | \( 1 + 15 T + 303 p^{2} T^{2} + 5553 p^{2} T^{3} + 29201 p^{5} T^{4} + 252033 p^{6} T^{5} + 29201 p^{14} T^{6} + 5553 p^{20} T^{7} + 303 p^{29} T^{8} + 15 p^{36} T^{9} + p^{45} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 + 1803 T + 5673036 T^{2} + 2122118037 p T^{3} + 154127491547 p^{3} T^{4} + 217433194071432 p^{3} T^{5} + 154127491547 p^{12} T^{6} + 2122118037 p^{19} T^{7} + 5673036 p^{27} T^{8} + 1803 p^{36} T^{9} + p^{45} T^{10} \) | |
| 7 | $C_2 \wr S_5$ | \( 1 - 10099 T + 139254174 T^{2} - 122453891247 p T^{3} + 166521050648013 p^{2} T^{4} - 122374113130618344 p^{3} T^{5} + 166521050648013 p^{11} T^{6} - 122453891247 p^{19} T^{7} + 139254174 p^{27} T^{8} - 10099 p^{36} T^{9} + p^{45} T^{10} \) | |
| 11 | $C_2 \wr S_5$ | \( 1 + 121746 T + 14269408587 T^{2} + 1047597531024144 T^{3} + 70977489220411684558 T^{4} + \)\(36\!\cdots\!52\)\( T^{5} + 70977489220411684558 p^{9} T^{6} + 1047597531024144 p^{18} T^{7} + 14269408587 p^{27} T^{8} + 121746 p^{36} T^{9} + p^{45} T^{10} \) | |
| 17 | $C_2 \wr S_5$ | \( 1 - 29157 p T + 303748065600 T^{2} - 136238774144131095 T^{3} + \)\(32\!\cdots\!83\)\( p T^{4} - \)\(62\!\cdots\!64\)\( p^{2} T^{5} + \)\(32\!\cdots\!83\)\( p^{10} T^{6} - 136238774144131095 p^{18} T^{7} + 303748065600 p^{27} T^{8} - 29157 p^{37} T^{9} + p^{45} T^{10} \) | |
| 19 | $C_2 \wr S_5$ | \( 1 + 840738 T + 1352453971283 T^{2} + 953145467206471696 T^{3} + \)\(83\!\cdots\!34\)\( T^{4} + \)\(43\!\cdots\!52\)\( T^{5} + \)\(83\!\cdots\!34\)\( p^{9} T^{6} + 953145467206471696 p^{18} T^{7} + 1352453971283 p^{27} T^{8} + 840738 p^{36} T^{9} + p^{45} T^{10} \) | |
| 23 | $C_2 \wr S_5$ | \( 1 - 592152 T + 5512736304883 T^{2} + 937525593298362720 T^{3} + \)\(10\!\cdots\!94\)\( T^{4} + \)\(33\!\cdots\!76\)\( p T^{5} + \)\(10\!\cdots\!94\)\( p^{9} T^{6} + 937525593298362720 p^{18} T^{7} + 5512736304883 p^{27} T^{8} - 592152 p^{36} T^{9} + p^{45} T^{10} \) | |
| 29 | $C_2 \wr S_5$ | \( 1 + 10678182 T + 88796663366937 T^{2} + \)\(54\!\cdots\!56\)\( T^{3} + \)\(28\!\cdots\!78\)\( T^{4} + \)\(11\!\cdots\!44\)\( T^{5} + \)\(28\!\cdots\!78\)\( p^{9} T^{6} + \)\(54\!\cdots\!56\)\( p^{18} T^{7} + 88796663366937 p^{27} T^{8} + 10678182 p^{36} T^{9} + p^{45} T^{10} \) | |
| 31 | $C_2 \wr S_5$ | \( 1 - 12885296 T + 164536395807867 T^{2} - \)\(12\!\cdots\!24\)\( T^{3} + \)\(92\!\cdots\!78\)\( T^{4} - \)\(48\!\cdots\!72\)\( T^{5} + \)\(92\!\cdots\!78\)\( p^{9} T^{6} - \)\(12\!\cdots\!24\)\( p^{18} T^{7} + 164536395807867 p^{27} T^{8} - 12885296 p^{36} T^{9} + p^{45} T^{10} \) | |
| 37 | $C_2 \wr S_5$ | \( 1 - 7171823 T + 354892872282468 T^{2} - \)\(10\!\cdots\!13\)\( T^{3} + \)\(60\!\cdots\!79\)\( T^{4} - \)\(91\!\cdots\!16\)\( T^{5} + \)\(60\!\cdots\!79\)\( p^{9} T^{6} - \)\(10\!\cdots\!13\)\( p^{18} T^{7} + 354892872282468 p^{27} T^{8} - 7171823 p^{36} T^{9} + p^{45} T^{10} \) | |
| 41 | $C_2 \wr S_5$ | \( 1 + 9294012 T + 571994210722321 T^{2} + \)\(89\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!02\)\( T^{4} + \)\(34\!\cdots\!04\)\( T^{5} + \)\(16\!\cdots\!02\)\( p^{9} T^{6} + \)\(89\!\cdots\!60\)\( p^{18} T^{7} + 571994210722321 p^{27} T^{8} + 9294012 p^{36} T^{9} + p^{45} T^{10} \) | |
| 43 | $C_2 \wr S_5$ | \( 1 - 12831975 T + 322694905923638 T^{2} - \)\(26\!\cdots\!89\)\( T^{3} + \)\(62\!\cdots\!77\)\( T^{4} - \)\(43\!\cdots\!96\)\( T^{5} + \)\(62\!\cdots\!77\)\( p^{9} T^{6} - \)\(26\!\cdots\!89\)\( p^{18} T^{7} + 322694905923638 p^{27} T^{8} - 12831975 p^{36} T^{9} + p^{45} T^{10} \) | |
| 47 | $C_2 \wr S_5$ | \( 1 + 43354215 T + 2527109677831630 T^{2} + \)\(75\!\cdots\!93\)\( T^{3} + \)\(33\!\cdots\!41\)\( T^{4} + \)\(71\!\cdots\!56\)\( T^{5} + \)\(33\!\cdots\!41\)\( p^{9} T^{6} + \)\(75\!\cdots\!93\)\( p^{18} T^{7} + 2527109677831630 p^{27} T^{8} + 43354215 p^{36} T^{9} + p^{45} T^{10} \) | |
| 53 | $C_2 \wr S_5$ | \( 1 + 93231780 T + 14353270896409645 T^{2} + \)\(11\!\cdots\!96\)\( T^{3} + \)\(88\!\cdots\!06\)\( T^{4} + \)\(52\!\cdots\!08\)\( T^{5} + \)\(88\!\cdots\!06\)\( p^{9} T^{6} + \)\(11\!\cdots\!96\)\( p^{18} T^{7} + 14353270896409645 p^{27} T^{8} + 93231780 p^{36} T^{9} + p^{45} T^{10} \) | |
| 59 | $C_2 \wr S_5$ | \( 1 + 246496182 T + 33223507046890507 T^{2} + \)\(26\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!50\)\( T^{4} + \)\(12\!\cdots\!36\)\( T^{5} + \)\(17\!\cdots\!50\)\( p^{9} T^{6} + \)\(26\!\cdots\!60\)\( p^{18} T^{7} + 33223507046890507 p^{27} T^{8} + 246496182 p^{36} T^{9} + p^{45} T^{10} \) | |
| 61 | $C_2 \wr S_5$ | \( 1 + 132232612 T + 51932556467342037 T^{2} + \)\(51\!\cdots\!12\)\( T^{3} + \)\(11\!\cdots\!74\)\( T^{4} + \)\(84\!\cdots\!48\)\( T^{5} + \)\(11\!\cdots\!74\)\( p^{9} T^{6} + \)\(51\!\cdots\!12\)\( p^{18} T^{7} + 51932556467342037 p^{27} T^{8} + 132232612 p^{36} T^{9} + p^{45} T^{10} \) | |
| 67 | $C_2 \wr S_5$ | \( 1 + 369388534 T + 167732159711342355 T^{2} + \)\(38\!\cdots\!64\)\( T^{3} + \)\(97\!\cdots\!58\)\( T^{4} + \)\(15\!\cdots\!40\)\( T^{5} + \)\(97\!\cdots\!58\)\( p^{9} T^{6} + \)\(38\!\cdots\!64\)\( p^{18} T^{7} + 167732159711342355 p^{27} T^{8} + 369388534 p^{36} T^{9} + p^{45} T^{10} \) | |
| 71 | $C_2 \wr S_5$ | \( 1 + 212150457 T + 126078700918544326 T^{2} + \)\(34\!\cdots\!35\)\( T^{3} + \)\(83\!\cdots\!89\)\( T^{4} + \)\(22\!\cdots\!08\)\( T^{5} + \)\(83\!\cdots\!89\)\( p^{9} T^{6} + \)\(34\!\cdots\!35\)\( p^{18} T^{7} + 126078700918544326 p^{27} T^{8} + 212150457 p^{36} T^{9} + p^{45} T^{10} \) | |
| 73 | $C_2 \wr S_5$ | \( 1 + 252729806 T + 113432726759207253 T^{2} + \)\(15\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!58\)\( T^{4} + \)\(17\!\cdots\!28\)\( T^{5} + \)\(10\!\cdots\!58\)\( p^{9} T^{6} + \)\(15\!\cdots\!40\)\( p^{18} T^{7} + 113432726759207253 p^{27} T^{8} + 252729806 p^{36} T^{9} + p^{45} T^{10} \) | |
| 79 | $C_2 \wr S_5$ | \( 1 + 1247271728 T + 1135234985129373579 T^{2} + \)\(68\!\cdots\!96\)\( T^{3} + \)\(33\!\cdots\!94\)\( T^{4} + \)\(12\!\cdots\!92\)\( T^{5} + \)\(33\!\cdots\!94\)\( p^{9} T^{6} + \)\(68\!\cdots\!96\)\( p^{18} T^{7} + 1135234985129373579 p^{27} T^{8} + 1247271728 p^{36} T^{9} + p^{45} T^{10} \) | |
| 83 | $C_2 \wr S_5$ | \( 1 + 1696894296 T + 1900005505469354847 T^{2} + \)\(14\!\cdots\!20\)\( T^{3} + \)\(90\!\cdots\!82\)\( T^{4} + \)\(43\!\cdots\!16\)\( T^{5} + \)\(90\!\cdots\!82\)\( p^{9} T^{6} + \)\(14\!\cdots\!20\)\( p^{18} T^{7} + 1900005505469354847 p^{27} T^{8} + 1696894296 p^{36} T^{9} + p^{45} T^{10} \) | |
| 89 | $C_2 \wr S_5$ | \( 1 - 753854382 T + 1391064645589335141 T^{2} - \)\(88\!\cdots\!08\)\( T^{3} + \)\(90\!\cdots\!06\)\( T^{4} - \)\(43\!\cdots\!00\)\( T^{5} + \)\(90\!\cdots\!06\)\( p^{9} T^{6} - \)\(88\!\cdots\!08\)\( p^{18} T^{7} + 1391064645589335141 p^{27} T^{8} - 753854382 p^{36} T^{9} + p^{45} T^{10} \) | |
| 97 | $C_2 \wr S_5$ | \( 1 - 3824606 T + 1359157730156522205 T^{2} + \)\(14\!\cdots\!24\)\( T^{3} + \)\(16\!\cdots\!14\)\( T^{4} + \)\(77\!\cdots\!52\)\( T^{5} + \)\(16\!\cdots\!14\)\( p^{9} T^{6} + \)\(14\!\cdots\!24\)\( p^{18} T^{7} + 1359157730156522205 p^{27} T^{8} - 3824606 p^{36} T^{9} + p^{45} T^{10} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.34479932770188704101614407882, −7.32048041366611808641467211954, −7.24474218217664709074168206206, −6.43146944984247054531534436405, −6.20712097586951832241324840627, −6.05944702904612976942238715271, −5.73503965023680589402076492967, −5.69805649350041194026837434644, −5.27334035077340168717428421419, −5.07185312068360161165344685006, −4.66760598741898621022299269417, −4.50647220648620523649038054615, −4.33549415577205195066128810935, −4.27100463854319484887376373174, −4.12075583347254689604654219873, −3.26511657795031569860148068458, −3.22366791931610529478036841669, −3.08144236756302089480049040464, −3.07217613623678968420514448092, −2.25134408688810868612373012497, −1.92430050765043199594857755779, −1.79641490188599927354328740065, −1.33503831324259175371003039560, −1.13278098154286283093504415550, −0.946924238943557130341681988103, 0, 0, 0, 0, 0, 0.946924238943557130341681988103, 1.13278098154286283093504415550, 1.33503831324259175371003039560, 1.79641490188599927354328740065, 1.92430050765043199594857755779, 2.25134408688810868612373012497, 3.07217613623678968420514448092, 3.08144236756302089480049040464, 3.22366791931610529478036841669, 3.26511657795031569860148068458, 4.12075583347254689604654219873, 4.27100463854319484887376373174, 4.33549415577205195066128810935, 4.50647220648620523649038054615, 4.66760598741898621022299269417, 5.07185312068360161165344685006, 5.27334035077340168717428421419, 5.69805649350041194026837434644, 5.73503965023680589402076492967, 6.05944702904612976942238715271, 6.20712097586951832241324840627, 6.43146944984247054531534436405, 7.24474218217664709074168206206, 7.32048041366611808641467211954, 7.34479932770188704101614407882