Dirichlet series
| L(s) = 1 | − 161·3-s + 1.80e3·5-s − 1.00e4·7-s − 5.71e3·9-s − 1.21e5·11-s + 1.42e5·13-s − 2.90e5·15-s − 4.95e5·17-s + 8.40e5·19-s + 1.62e6·21-s + 5.92e5·23-s − 2.42e6·25-s + 1.08e6·27-s + 1.06e7·29-s − 1.28e7·31-s + 1.96e7·33-s − 1.82e7·35-s + 7.17e6·37-s − 2.29e7·39-s + 9.29e6·41-s − 1.28e7·43-s − 1.03e7·45-s − 4.33e7·47-s − 3.72e7·49-s + 7.98e7·51-s + 9.32e7·53-s − 2.19e8·55-s + ⋯ |
| L(s) = 1 | − 1.14·3-s + 1.29·5-s − 1.58·7-s − 0.290·9-s − 2.50·11-s + 1.38·13-s − 1.48·15-s − 1.43·17-s + 1.48·19-s + 1.82·21-s + 0.441·23-s − 1.24·25-s + 0.392·27-s + 2.80·29-s − 2.50·31-s + 2.87·33-s − 2.05·35-s + 0.629·37-s − 1.59·39-s + 0.513·41-s − 0.572·43-s − 0.374·45-s − 1.29·47-s − 0.923·49-s + 1.65·51-s + 1.62·53-s − 3.23·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(10\) |
| Conductor: | \(2^{20} \cdot 13^{5}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.41092\times 10^{10}\) |
| Root analytic conductor: | \(10.3502\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(5\) |
| Selberg data: | \((10,\ 2^{20} \cdot 13^{5} ,\ ( \ : 9/2, 9/2, 9/2, 9/2, 9/2 ),\ -1 )\) |
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 | 2 | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 - p^{4} T )^{5} \) | |
| good | 3 | $C_2 \wr S_5$ | \( 1 + 161 T + 10546 p T^{2} + 60851 p^{4} T^{3} + 37844987 p^{3} T^{4} + 1621483792 p^{4} T^{5} + 37844987 p^{12} T^{6} + 60851 p^{22} T^{7} + 10546 p^{28} T^{8} + 161 p^{36} T^{9} + p^{45} T^{10} \) |
| 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
−6.36462749398838407803608042422, −6.11263202233780018088132064070, −6.11085580338786371832332775452, −6.05247961861548853021674216887, −5.90425844863114287575812582517, −5.33480214382167014845621576701, −5.30524403864015557455964673795, −5.23371043423504310761354920702, −5.08359343174844389881588039583, −4.61478274580712744175322330520, −4.53054225710761360075774885240, −3.99912029286932311295399467493, −3.75829384685768785142009366180, −3.70088849799796462963140336257, −3.38678278814354224273486520182, −2.82386876762059450946972992543, −2.74414690373965449776635639408, −2.72932067970497987323170898050, −2.66579865352240529894891561831, −2.16573680279845600868060409414, −1.73102366334286269743387912110, −1.58865825813432691043483875373, −1.26658690714945075218574020483, −1.13933177173345191038160784256, −0.816540532126358517243907746039, 0, 0, 0, 0, 0, 0.816540532126358517243907746039, 1.13933177173345191038160784256, 1.26658690714945075218574020483, 1.58865825813432691043483875373, 1.73102366334286269743387912110, 2.16573680279845600868060409414, 2.66579865352240529894891561831, 2.72932067970497987323170898050, 2.74414690373965449776635639408, 2.82386876762059450946972992543, 3.38678278814354224273486520182, 3.70088849799796462963140336257, 3.75829384685768785142009366180, 3.99912029286932311295399467493, 4.53054225710761360075774885240, 4.61478274580712744175322330520, 5.08359343174844389881588039583, 5.23371043423504310761354920702, 5.30524403864015557455964673795, 5.33480214382167014845621576701, 5.90425844863114287575812582517, 6.05247961861548853021674216887, 6.11085580338786371832332775452, 6.11263202233780018088132064070, 6.36462749398838407803608042422