Dirichlet series
| L(s) = 1 | − 12·3-s + 750·5-s − 2.05e3·7-s − 3.26e3·9-s − 4.84e3·11-s − 5.20e3·13-s − 9.00e3·15-s − 2.97e3·17-s − 6.88e4·19-s + 2.46e4·21-s − 3.57e4·23-s + 3.28e5·25-s − 4.85e3·27-s + 6.66e4·29-s − 1.84e5·31-s + 5.81e4·33-s − 1.54e6·35-s − 5.57e4·37-s + 6.24e4·39-s + 6.61e5·41-s − 1.36e6·43-s − 2.44e6·45-s − 6.00e4·47-s + 2.47e6·49-s + 3.57e4·51-s − 1.48e6·53-s − 3.63e6·55-s + ⋯ |
| L(s) = 1 | − 0.256·3-s + 2.68·5-s − 2.26·7-s − 1.49·9-s − 1.09·11-s − 0.656·13-s − 0.688·15-s − 0.146·17-s − 2.30·19-s + 0.581·21-s − 0.613·23-s + 21/5·25-s − 0.0474·27-s + 0.507·29-s − 1.11·31-s + 0.281·33-s − 6.08·35-s − 0.180·37-s + 0.168·39-s + 1.49·41-s − 2.62·43-s − 4.00·45-s − 0.0843·47-s + 3·49-s + 0.0376·51-s − 1.36·53-s − 2.94·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{18} \cdot 5^{6} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.47805\times 10^{11}\) |
| Root analytic conductor: | \(9.35242\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{18} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - p^{3} T )^{6} \) | |
| 7 | \( ( 1 + p^{3} T )^{6} \) | |
| good | 3 | \( 1 + 4 p T + 1136 p T^{2} + 84920 T^{3} + 2409592 p T^{4} + 26929540 p^{2} T^{5} + 382113074 p^{3} T^{6} + 26929540 p^{9} T^{7} + 2409592 p^{15} T^{8} + 84920 p^{21} T^{9} + 1136 p^{29} T^{10} + 4 p^{36} T^{11} + p^{42} T^{12} \) |
| 11 | \( 1 + 4844 T + 78473368 T^{2} + 233890722744 T^{3} + 2607097751706304 T^{4} + 5653834339801255860 T^{5} + \)\(58\!\cdots\!34\)\( T^{6} + 5653834339801255860 p^{7} T^{7} + 2607097751706304 p^{14} T^{8} + 233890722744 p^{21} T^{9} + 78473368 p^{28} T^{10} + 4844 p^{35} T^{11} + p^{42} T^{12} \) | |
| 13 | \( 1 + 400 p T + 256351728 T^{2} + 784640424836 T^{3} + 28606245516800976 T^{4} + 49435823966319530032 T^{5} + \)\(20\!\cdots\!94\)\( T^{6} + 49435823966319530032 p^{7} T^{7} + 28606245516800976 p^{14} T^{8} + 784640424836 p^{21} T^{9} + 256351728 p^{28} T^{10} + 400 p^{36} T^{11} + p^{42} T^{12} \) | |
| 17 | \( 1 + 2976 T + 917377272 T^{2} + 2508152510764 T^{3} + 750769236339676104 T^{4} + \)\(16\!\cdots\!92\)\( T^{5} + \)\(32\!\cdots\!02\)\( T^{6} + \)\(16\!\cdots\!92\)\( p^{7} T^{7} + 750769236339676104 p^{14} T^{8} + 2508152510764 p^{21} T^{9} + 917377272 p^{28} T^{10} + 2976 p^{35} T^{11} + p^{42} T^{12} \) | |
| 19 | \( 1 + 68872 T + 3881282890 T^{2} + 148122138420856 T^{3} + 6273335362454508135 T^{4} + \)\(21\!\cdots\!84\)\( T^{5} + \)\(72\!\cdots\!72\)\( T^{6} + \)\(21\!\cdots\!84\)\( p^{7} T^{7} + 6273335362454508135 p^{14} T^{8} + 148122138420856 p^{21} T^{9} + 3881282890 p^{28} T^{10} + 68872 p^{35} T^{11} + p^{42} T^{12} \) | |
| 23 | \( 1 + 35776 T + 9937206562 T^{2} + 246155827095168 T^{3} + 59712390012487766479 T^{4} + \)\(10\!\cdots\!88\)\( T^{5} + \)\(22\!\cdots\!72\)\( T^{6} + \)\(10\!\cdots\!88\)\( p^{7} T^{7} + 59712390012487766479 p^{14} T^{8} + 246155827095168 p^{21} T^{9} + 9937206562 p^{28} T^{10} + 35776 p^{35} T^{11} + p^{42} T^{12} \) | |
| 29 | \( 1 - 66696 T + 46999250712 T^{2} - 3250551423222492 T^{3} + \)\(12\!\cdots\!92\)\( T^{4} - \)\(94\!\cdots\!44\)\( T^{5} + \)\(26\!\cdots\!94\)\( T^{6} - \)\(94\!\cdots\!44\)\( p^{7} T^{7} + \)\(12\!\cdots\!92\)\( p^{14} T^{8} - 3250551423222492 p^{21} T^{9} + 46999250712 p^{28} T^{10} - 66696 p^{35} T^{11} + p^{42} T^{12} \) | |
| 31 | \( 1 + 184584 T + 47083124010 T^{2} + 7414102371848152 T^{3} + \)\(11\!\cdots\!79\)\( T^{4} + \)\(18\!\cdots\!76\)\( T^{5} + \)\(27\!\cdots\!80\)\( T^{6} + \)\(18\!\cdots\!76\)\( p^{7} T^{7} + \)\(11\!\cdots\!79\)\( p^{14} T^{8} + 7414102371848152 p^{21} T^{9} + 47083124010 p^{28} T^{10} + 184584 p^{35} T^{11} + p^{42} T^{12} \) | |
| 37 | \( 1 + 55716 T + 102979802346 T^{2} - 15266724753732476 T^{3} + \)\(19\!\cdots\!15\)\( T^{4} - \)\(43\!\cdots\!28\)\( p T^{5} + \)\(15\!\cdots\!68\)\( T^{6} - \)\(43\!\cdots\!28\)\( p^{8} T^{7} + \)\(19\!\cdots\!15\)\( p^{14} T^{8} - 15266724753732476 p^{21} T^{9} + 102979802346 p^{28} T^{10} + 55716 p^{35} T^{11} + p^{42} T^{12} \) | |
| 41 | \( 1 - 661852 T + 925927436426 T^{2} - 511321847807971660 T^{3} + \)\(40\!\cdots\!31\)\( T^{4} - \)\(17\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!24\)\( T^{6} - \)\(17\!\cdots\!68\)\( p^{7} T^{7} + \)\(40\!\cdots\!31\)\( p^{14} T^{8} - 511321847807971660 p^{21} T^{9} + 925927436426 p^{28} T^{10} - 661852 p^{35} T^{11} + p^{42} T^{12} \) | |
| 43 | \( 1 + 1367576 T + 1698547958394 T^{2} + 1303427892125267752 T^{3} + \)\(93\!\cdots\!39\)\( T^{4} + \)\(52\!\cdots\!88\)\( T^{5} + \)\(29\!\cdots\!52\)\( T^{6} + \)\(52\!\cdots\!88\)\( p^{7} T^{7} + \)\(93\!\cdots\!39\)\( p^{14} T^{8} + 1303427892125267752 p^{21} T^{9} + 1698547958394 p^{28} T^{10} + 1367576 p^{35} T^{11} + p^{42} T^{12} \) | |
| 47 | \( 1 + 60036 T + 1076704030776 T^{2} + 121328803271064480 T^{3} + \)\(90\!\cdots\!68\)\( T^{4} + \)\(15\!\cdots\!56\)\( T^{5} + \)\(48\!\cdots\!50\)\( T^{6} + \)\(15\!\cdots\!56\)\( p^{7} T^{7} + \)\(90\!\cdots\!68\)\( p^{14} T^{8} + 121328803271064480 p^{21} T^{9} + 1076704030776 p^{28} T^{10} + 60036 p^{35} T^{11} + p^{42} T^{12} \) | |
| 53 | \( 1 + 1484068 T + 2188089874418 T^{2} + 2026868253977007460 T^{3} + \)\(34\!\cdots\!63\)\( T^{4} + \)\(30\!\cdots\!44\)\( T^{5} + \)\(41\!\cdots\!40\)\( T^{6} + \)\(30\!\cdots\!44\)\( p^{7} T^{7} + \)\(34\!\cdots\!63\)\( p^{14} T^{8} + 2026868253977007460 p^{21} T^{9} + 2188089874418 p^{28} T^{10} + 1484068 p^{35} T^{11} + p^{42} T^{12} \) | |
| 59 | \( 1 - 859224 T + 8876158854498 T^{2} - 9008604849486708136 T^{3} + \)\(43\!\cdots\!11\)\( T^{4} - \)\(38\!\cdots\!28\)\( T^{5} + \)\(13\!\cdots\!32\)\( T^{6} - \)\(38\!\cdots\!28\)\( p^{7} T^{7} + \)\(43\!\cdots\!11\)\( p^{14} T^{8} - 9008604849486708136 p^{21} T^{9} + 8876158854498 p^{28} T^{10} - 859224 p^{35} T^{11} + p^{42} T^{12} \) | |
| 61 | \( 1 - 2632548 T + 14301167557746 T^{2} - 18916583756342066340 T^{3} + \)\(64\!\cdots\!75\)\( T^{4} - \)\(38\!\cdots\!48\)\( T^{5} + \)\(18\!\cdots\!60\)\( T^{6} - \)\(38\!\cdots\!48\)\( p^{7} T^{7} + \)\(64\!\cdots\!75\)\( p^{14} T^{8} - 18916583756342066340 p^{21} T^{9} + 14301167557746 p^{28} T^{10} - 2632548 p^{35} T^{11} + p^{42} T^{12} \) | |
| 67 | \( 1 + 1354376 T + 15927150921682 T^{2} + 21083884322727449912 T^{3} + \)\(13\!\cdots\!87\)\( T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(85\!\cdots\!36\)\( T^{6} + \)\(10\!\cdots\!44\)\( p^{7} T^{7} + \)\(13\!\cdots\!87\)\( p^{14} T^{8} + 21083884322727449912 p^{21} T^{9} + 15927150921682 p^{28} T^{10} + 1354376 p^{35} T^{11} + p^{42} T^{12} \) | |
| 71 | \( 1 - 4395472 T + 35357547199498 T^{2} - \)\(14\!\cdots\!96\)\( T^{3} + \)\(59\!\cdots\!11\)\( T^{4} - \)\(23\!\cdots\!68\)\( T^{5} + \)\(65\!\cdots\!68\)\( T^{6} - \)\(23\!\cdots\!68\)\( p^{7} T^{7} + \)\(59\!\cdots\!11\)\( p^{14} T^{8} - \)\(14\!\cdots\!96\)\( p^{21} T^{9} + 35357547199498 p^{28} T^{10} - 4395472 p^{35} T^{11} + p^{42} T^{12} \) | |
| 73 | \( 1 + 3842084 T + 35794063603410 T^{2} + \)\(10\!\cdots\!32\)\( T^{3} + \)\(73\!\cdots\!63\)\( T^{4} + \)\(18\!\cdots\!68\)\( T^{5} + \)\(99\!\cdots\!08\)\( T^{6} + \)\(18\!\cdots\!68\)\( p^{7} T^{7} + \)\(73\!\cdots\!63\)\( p^{14} T^{8} + \)\(10\!\cdots\!32\)\( p^{21} T^{9} + 35794063603410 p^{28} T^{10} + 3842084 p^{35} T^{11} + p^{42} T^{12} \) | |
| 79 | \( 1 + 2298668 T + 72995843074864 T^{2} + 92034708759691066064 T^{3} + \)\(23\!\cdots\!80\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(48\!\cdots\!78\)\( T^{6} + \)\(12\!\cdots\!00\)\( p^{7} T^{7} + \)\(23\!\cdots\!80\)\( p^{14} T^{8} + 92034708759691066064 p^{21} T^{9} + 72995843074864 p^{28} T^{10} + 2298668 p^{35} T^{11} + p^{42} T^{12} \) | |
| 83 | \( 1 + 9094248 T + 81842080927314 T^{2} + \)\(32\!\cdots\!48\)\( T^{3} + \)\(96\!\cdots\!35\)\( T^{4} - \)\(49\!\cdots\!48\)\( T^{5} - \)\(27\!\cdots\!08\)\( T^{6} - \)\(49\!\cdots\!48\)\( p^{7} T^{7} + \)\(96\!\cdots\!35\)\( p^{14} T^{8} + \)\(32\!\cdots\!48\)\( p^{21} T^{9} + 81842080927314 p^{28} T^{10} + 9094248 p^{35} T^{11} + p^{42} T^{12} \) | |
| 89 | \( 1 + 11050996 T + 200402878956714 T^{2} + \)\(17\!\cdots\!88\)\( T^{3} + \)\(18\!\cdots\!95\)\( T^{4} + \)\(13\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!04\)\( T^{6} + \)\(13\!\cdots\!32\)\( p^{7} T^{7} + \)\(18\!\cdots\!95\)\( p^{14} T^{8} + \)\(17\!\cdots\!88\)\( p^{21} T^{9} + 200402878956714 p^{28} T^{10} + 11050996 p^{35} T^{11} + p^{42} T^{12} \) | |
| 97 | \( 1 + 27508112 T + 556247329631208 T^{2} + \)\(75\!\cdots\!96\)\( T^{3} + \)\(91\!\cdots\!48\)\( T^{4} + \)\(89\!\cdots\!28\)\( T^{5} + \)\(85\!\cdots\!22\)\( T^{6} + \)\(89\!\cdots\!28\)\( p^{7} T^{7} + \)\(91\!\cdots\!48\)\( p^{14} T^{8} + \)\(75\!\cdots\!96\)\( p^{21} T^{9} + 556247329631208 p^{28} T^{10} + 27508112 p^{35} T^{11} + p^{42} T^{12} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.95471325423613105299173976786, −5.44972404992896829613184875605, −5.27286477097354273496858672897, −5.25699622607307559957876563564, −5.24344740259029325495820262567, −5.04740048903626019872285278093, −4.81958345983008816479698223175, −4.41154365598889111234610393131, −4.12858719986819188256566393346, −3.88674411457763463797124044311, −3.76716897856562518114624067226, −3.72743315743334950757365968271, −3.58132392410448886393242585274, −2.82225802994738312777505683317, −2.73281900830335278965642373403, −2.70560608459407610966803980858, −2.63365974665963942942121365001, −2.54722588468880626480739262221, −2.47067084813096060281292102311, −2.03759967544455217954654853710, −1.60199119163303608668830026678, −1.59572941527179191058678881983, −1.27285473753501544811479708293, −1.16516650525737248685785155315, −0.989057191557874156875549003466, 0, 0, 0, 0, 0, 0, 0.989057191557874156875549003466, 1.16516650525737248685785155315, 1.27285473753501544811479708293, 1.59572941527179191058678881983, 1.60199119163303608668830026678, 2.03759967544455217954654853710, 2.47067084813096060281292102311, 2.54722588468880626480739262221, 2.63365974665963942942121365001, 2.70560608459407610966803980858, 2.73281900830335278965642373403, 2.82225802994738312777505683317, 3.58132392410448886393242585274, 3.72743315743334950757365968271, 3.76716897856562518114624067226, 3.88674411457763463797124044311, 4.12858719986819188256566393346, 4.41154365598889111234610393131, 4.81958345983008816479698223175, 5.04740048903626019872285278093, 5.24344740259029325495820262567, 5.25699622607307559957876563564, 5.27286477097354273496858672897, 5.44972404992896829613184875605, 5.95471325423613105299173976786
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.